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Theorem onsseli 5811
Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
Hypotheses
Ref Expression
on.1 𝐴 ∈ On
on.2 𝐵 ∈ On
Assertion
Ref Expression
onsseli (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem onsseli
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 on.2 . 2 𝐵 ∈ On
3 onsseleq 5734 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3mp2an 707 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383   = wceq 1480  wcel 1987  wss 3560  Oncon0 5692
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 4751  ax-nul 4759  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696
This theorem is referenced by:  cardom  8772  tskcard  9563
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