Theorem on0eqel 6488

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

Step	Hyp	Ref	Expression
1		0ss 4396	. . 3 ⊢ ∅ ⊆ 𝐴
2		0elon 6418	. . . 4 ⊢ ∅ ∈ On
3		onsseleq 6405	. . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
4	2, 3	mpan 688	. . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
5	1, 4	mpbii 232	. 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6		eqcom 2739	. . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅)
7	6	orbi2i 911	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅))
8		orcom 868	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
9	7, 8	bitri 274	. 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
10	5, 9	sylib 217	1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948  ∅c0 4322  Oncon0 6364

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368

This theorem is referenced by:  snsn0non  6489  onxpdisj  6490  omabs  8652  cnfcom3lem  9700  0elold  27628  onexlimgt  42294  onexoegt  42295  oe0rif  42337  oege1  42358  onmcl  42383  omabs2  42384  omcl2  42385