Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssext Structured version   Visualization version   GIF version

Theorem ssext 4884
 Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
Assertion
Ref Expression
ssext (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssext
StepHypRef Expression
1 ssextss 4883 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 ssextss 4883 . . 3 (𝐵𝐴 ↔ ∀𝑥(𝑥𝐵𝑥𝐴))
31, 2anbi12i 732 . 2 ((𝐴𝐵𝐵𝐴) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐵𝑥𝐴)))
4 eqss 3598 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 albiim 1813 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐵𝑥𝐴)))
63, 4, 53bitr4i 292 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   = wceq 1480   ⊆ wss 3555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator