Theorem on0eqel 5845

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 3972	. . 3 ⊢ ∅ ⊆ 𝐴
2		0elon 5778	. . . 4 ⊢ ∅ ∈ On
3		onsseleq 5765	. . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
4	2, 3	mpan 706	. . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
5	1, 4	mpbii 223	. 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6		eqcom 2629	. . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅)
7	6	orbi2i 541	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅))
8		orcom 402	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
9	7, 8	bitri 264	. 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
10	5, 9	sylib 208	1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  snsn0non  5846  onxpdisj  5847  omabs  7727  cnfcom3lem  8600