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Theorem onsseli 6004
 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 5927 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3mp2an 710 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 382   = wceq 1632   ∈ wcel 2140   ⊆ wss 3716  Oncon0 5885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-ord 5888  df-on 5889 This theorem is referenced by:  cardom  9023  tskcard  9816
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