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Theorem sbthbg 4438
Description: Schroeder-Bernstein Theorem and its converse.
Assertion
Ref Expression
sbthbg |- (B e. C -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
Proof of Theorem sbthbg
StepHypRef Expression
1 sbth 4437 . 2 |- ((A ~<_ B /\ B ~<_ A) -> A ~~ B)
2 endom 4366 . . . 4 |- (A ~~ B -> A ~<_ B)
32a1i 8 . . 3 |- (B e. C -> (A ~~ B -> A ~<_ B))
4 ensymg 4392 . . . 4 |- (B e. C -> (A ~~ B -> B ~~ A))
5 endom 4366 . . . 4 |- (B ~~ A -> B ~<_ A)
64, 5syl6 22 . . 3 |- (B e. C -> (A ~~ B -> B ~<_ A))
73, 6jcad 598 . 2 |- (B e. C -> (A ~~ B -> (A ~<_ B /\ B ~<_ A)))
81, 7impbid2 516 1 |- (B e. C -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955   class class class wbr 2609   ~~ cen 4348   ~<_ cdom 4349
This theorem is referenced by:  sbthcl 4439  dom0 4445  xpen 4468  axgroth2 8717
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-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-er 4245  df-en 4351  df-dom 4352
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