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

Theorem on0eqel 5807
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 3949 . . 3 ∅ ⊆ 𝐴
2 0elon 5740 . . . 4 ∅ ∈ On
3 onsseleq 5727 . . . 4 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
42, 3mpan 705 . . 3 (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
51, 4mpbii 223 . 2 (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6 eqcom 2633 . . . 4 (∅ = 𝐴𝐴 = ∅)
76orbi2i 541 . . 3 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴𝐴 = ∅))
8 orcom 402 . . 3 ((∅ ∈ 𝐴𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
97, 8bitri 264 . 2 ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
105, 9sylib 208 1 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383   = wceq 1480  wcel 1992  wss 3560  c0 3896  Oncon0 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689
This theorem is referenced by:  snsn0non  5808  onxpdisj  5809  omabs  7673  cnfcom3lem  8545
  Copyright terms: Public domain W3C validator