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Theorem onsseli 6481
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 6401 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3mp2an 691 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 846   = wceq 1542  wcel 2107  wss 3946  Oncon0 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-tr 5264  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6363  df-on 6364
This theorem is referenced by:  cardom  9976  tskcard  10771
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