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Theorem unisn2 2866
Description: A version of unisn 2507 without the A e. V hypothesis. (Contributed by Stefan Allan, 14-Mar-06.)
Assertion
Ref Expression
unisn2 |- U.{A} e. {(/), A}
Proof of Theorem unisn2
StepHypRef Expression
1 unisng 2508 . . 3 |- (A e. V -> U.{A} = A)
2 eqid 1468 . . . . 5 |- A = A
32olci 271 . . . 4 |- (A = (/) \/ A = A)
4 elprg 2413 . . . 4 |- (A e. V -> (A e. {(/), A} <-> (A = (/) \/ A = A)))
53, 4mpbiri 194 . . 3 |- (A e. V -> A e. {(/), A})
61, 5eqeltrd 1540 . 2 |- (A e. V -> U.{A} e. {(/), A})
7 snprc 2433 . . . . 5 |- (-. A e. V <-> {A} = (/))
87biimp 151 . . . 4 |- (-. A e. V -> {A} = (/))
98unieqd 2502 . . 3 |- (-. A e. V -> U.{A} = U.(/))
10 uni0 2515 . . . 4 |- U.(/) = (/)
11 0ex 2701 . . . . 5 |- (/) e. V
1211pri1 2441 . . . 4 |- (/) e. {(/), A}
1310, 12eqeltr 1536 . . 3 |- U.(/) e. {(/), A}
149, 13syl6eqel 1548 . 2 |- (-. A e. V -> U.{A} e. {(/), A})
156, 14pm2.61i 126 1 |- U.{A} e. {(/), A}
Colors of variables: wff set class
Syntax hints:  -. wn 2   \/ wo 222   = wceq 953   e. wcel 955  Vcvv 1802  (/)c0 2270  {csn 2399  {cpr 2400  U.cuni 2493
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-10 963  ax-11 964  ax-12 965  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-nul 2700
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-sn 2402  df-pr 2403  df-uni 2494
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