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Theorem onsseli 6284
 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 6210 . 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 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ⊆ wss 3858  Oncon0 6169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-tr 5139  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-ord 6172  df-on 6173 This theorem is referenced by:  cardom  9448  tskcard  10241
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