Theorem on0eqel 6465

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 4353	. . 3 ⊢ ∅ ⊆ 𝐴
2		0elon 6395	. . . 4 ⊢ ∅ ∈ On
3		onsseleq 6381	. . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
4	2, 3	mpan 700	. . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
5	1, 4	mpbii 235	. 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6		eqcom 2768	. . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅)
7	6	orbi2i 923	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅))
8		orcom 881	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
9	7, 8	bitri 277	. 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
10	5, 9	sylib 220	1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 858   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904  ∅c0 4285  Oncon0 6340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6343  df-on 6344

This theorem is referenced by:  snsn0non  6466  onxpdisj  6467  omabs  8614  cnfcom3lem  9653  0elold  27978  onexlimgt  43773  onexoegt  43774  oe0rif  43815  oege1  43836  onmcl  43861  omabs2  43862  omcl2  43863