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Theorem on0eqel 6438
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 4354 . . 3 ∅ ⊆ 𝐴
2 0elon 6369 . . . 4 ∅ ∈ On
3 onsseleq 6356 . . . 4 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
42, 3mpan 688 . . 3 (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
51, 4mpbii 232 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6 eqcom 2743 . . . 4 (∅ = 𝐴𝐴 = ∅)
76orbi2i 911 . . 3 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴𝐴 = ∅))
8 orcom 868 . . 3 ((∅ ∈ 𝐴𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
97, 8bitri 274 . 2 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
105, 9sylib 217 1 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845   = wceq 1541  wcel 2106  wss 3908  c0 4280  Oncon0 6315
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-ord 6318  df-on 6319
This theorem is referenced by:  snsn0non  6439  onxpdisj  6440  omabs  8593  cnfcom3lem  9635  onexlimgt  41515  onexoegt  41516  oe0rif  41558  oege1  41578  omabs2  41603  omcl2  41604
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