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Theorem ssex 3857
 Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 3838 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
Hypothesis
Ref Expression
ssex.1 B V
Assertion
Ref Expression
ssex (ABA V)

Proof of Theorem ssex
StepHypRef Expression
1 df-ss 2899 . 2 (AB ↔ (AB) = A)
2 ssex.1 . . . 4 B V
32inex2 3855 . . 3 (AB) V
4 eleq1 2073 . . 3 ((AB) = A → ((AB) V ↔ A V))
53, 4mpbii 136 . 2 ((AB) = AA V)
61, 5sylbi 114 1 (ABA V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1223   ∈ wcel 1366  Vcvv 2526   ∩ cin 2884   ⊆ wss 2885 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838 This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-in 2892  df-ss 2899 This theorem is referenced by:  ssexi  3858  ssexg  3859  inteximm  3866  funimaexglem  4896  tfrexlem  5858  elinp  6314
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