Theorem on0eqel 6384

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 4330	. . 3 ⊢ ∅ ⊆ 𝐴
2		0elon 6319	. . . 4 ⊢ ∅ ∈ On
3		onsseleq 6307	. . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
4	2, 3	mpan 687	. . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
5	1, 4	mpbii 232	. 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6		eqcom 2745	. . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅)
7	6	orbi2i 910	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅))
8		orcom 867	. . 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 844   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∅c0 4256  Oncon0 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270

This theorem is referenced by:  snsn0non  6385  onxpdisj  6386  omabs  8481  cnfcom3lem  9461