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Theorem on0eqel 6286
Description: An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
Assertion
Ref Expression
on0eqel (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Proof of Theorem on0eqel
StepHypRef Expression
1 0ss 4322 . . 3 ∅ ⊆ 𝐴
2 0elon 6222 . . . 4 ∅ ∈ On
3 onsseleq 6210 . . . 4 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
42, 3mpan 689 . . 3 (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
51, 4mpbii 236 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6 eqcom 2829 . . . 4 (∅ = 𝐴𝐴 = ∅)
76orbi2i 910 . . 3 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴𝐴 = ∅))
8 orcom 867 . . 3 ((∅ ∈ 𝐴𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
97, 8bitri 278 . 2 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
105, 9sylib 221 1 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wo 844   = wceq 1538  wcel 2114  wss 3908  c0 4265  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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173
This theorem is referenced by:  snsn0non  6287  onxpdisj  6288  omabs  8261  cnfcom3lem  9154
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