MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  on0eqel Structured version   Visualization version   GIF version

Theorem on0eqel 5762
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 3924 . . 3 ∅ ⊆ 𝐴
2 0elon 5695 . . . 4 ∅ ∈ On
3 onsseleq 5682 . . . 4 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
42, 3mpan 702 . . 3 (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
51, 4mpbii 222 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6 eqcom 2617 . . . 4 (∅ = 𝐴𝐴 = ∅)
76orbi2i 540 . . 3 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴𝐴 = ∅))
8 orcom 401 . . 3 ((∅ ∈ 𝐴𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
97, 8bitri 263 . 2 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
105, 9sylib 207 1 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382   = wceq 1475  wcel 1977  wss 3540  c0 3874  Oncon0 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644
This theorem is referenced by:  snsn0non  5763  onxpdisj  5764  omabs  7614  cnfcom3lem  8483
  Copyright terms: Public domain W3C validator