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Theorem subvalt 5329
Description: Value of subtraction, which is the (unique) element x such that B + x = A. The notation U.{x e. CC | (B + x) = A} may at first seem cryptic but is actually a way of saying "the element x such that B + x = A" (see Theorem 8.17 of [Quine] p. 56); this works because there is only one such x as shown by negeu 5327, allowing us to exploit eusn 2436 and unisn 2507 (which you will find if you trace back the proof of subcl 5338).
Assertion
Ref Expression
subvalt |- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Distinct variable groups:   x,A   x,B
Proof of Theorem subvalt
StepHypRef Expression
1 axcnex 5239 . . . 4 |- CC e. V
21rabex 2715 . . 3 |- {x e. CC | (B + x) = A} e. V
32uniex 2861 . 2 |- U.{x e. CC | (B + x) = A} e. V
4 eqeq2 1476 . . . 4 |- (y = A -> ((z + x) = y <-> (z + x) = A))
54rabbisdv 1798 . . 3 |- (y = A -> {x e. CC | (z + x) = y} = {x e. CC | (z + x) = A})
65unieqd 2502 . 2 |- (y = A -> U.{x e. CC | (z + x) = y} = U.{x e. CC | (z + x) = A})
7 opreq1 3953 . . . . 5 |- (z = B -> (z + x) = (B + x))
87eqeq1d 1475 . . . 4 |- (z = B -> ((z + x) = A <-> (B + x) = A))
98rabbisdv 1798 . . 3 |- (z = B -> {x e. CC | (z + x) = A} = {x e. CC | (B + x) = A})
109unieqd 2502 . 2 |- (z = B -> U.{x e. CC | (z + x) = A} = U.{x e. CC | (B + x) = A})
11 df-sub 5328 . 2 |- - = {<.<.y, z>., w>. | ((y e. CC /\ z e. CC) /\ w = U.{x e. CC | (z + x) = y})}
123, 6, 10, 11oprabval2 4013 1 |- ((A e. CC /\ B e. CC) -> (A - B) = U.{x e. CC | (B + x) = A})
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {crab 1640  U.cuni 2493  (class class class)co 3948  CCcc 5204   + caddc 5209   - cmin 5264
This theorem is referenced by:  subcl 5338  subopr 5342  subadd 5343  addinv 8065
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-opr 3950  df-oprab 3951  df-qs 4250  df-ni 4972  df-nq 5010  df-np 5058  df-nr 5139  df-c 5212  df-sub 5328
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