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Theorem on0eqel 6507
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 4399 . . 3 ∅ ⊆ 𝐴
2 0elon 6437 . . . 4 ∅ ∈ On
3 onsseleq 6424 . . . 4 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
42, 3mpan 690 . . 3 (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
51, 4mpbii 233 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6 eqcom 2743 . . . 4 (∅ = 𝐴𝐴 = ∅)
76orbi2i 912 . . 3 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴𝐴 = ∅))
8 orcom 870 . . 3 ((∅ ∈ 𝐴𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
97, 8bitri 275 . 2 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
105, 9sylib 218 1 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847   = wceq 1539  wcel 2107  wss 3950  c0 4332  Oncon0 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387
This theorem is referenced by:  snsn0non  6508  onxpdisj  6509  omabs  8690  cnfcom3lem  9744  0elold  27948  onexlimgt  43260  onexoegt  43261  oe0rif  43303  oege1  43324  onmcl  43349  omabs2  43350  omcl2  43351
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