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

Theorem ordelpss 5893
 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 5892 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
2 df-pss 3739 . 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 382   ∈ wcel 2145   ≠ wne 2943   ⊆ wss 3723   ⊊ wpss 3724  Ord word 5864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868 This theorem is referenced by:  ordsseleq  5894  ordtri3or  5897  ordtr2  5910  onpsssuc  7170  php4  8307  nndomo  8314  ordpss  39180
 Copyright terms: Public domain W3C validator