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

Theorem ordelpss 5710
Description: For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
ordelpss ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))

Proof of Theorem ordelpss
StepHypRef Expression
1 ordelssne 5709 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
2 df-pss 3571 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
31, 2syl6bbr 278 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  wne 2790  wss 3555  wpss 3556  Ord word 5681
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-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685
This theorem is referenced by:  ordsseleq  5711  ordtri3or  5714  ordtr2  5727  onpsssuc  6966  php4  8091  nndomo  8098  ordpss  38137
  Copyright terms: Public domain W3C validator