Theorem on0eqel 6448

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 4340	. . 3 ⊢ ∅ ⊆ 𝐴
2		0elon 6378	. . . 4 ⊢ ∅ ∈ On
3		onsseleq 6364	. . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
4	2, 3	mpan 691	. . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴)))
5	1, 4	mpbii 233	. 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴))
6		eqcom 2743	. . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅)
7	6	orbi2i 913	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅))
8		orcom 871	. . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
9	7, 8	bitri 275	. 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
10	5, 9	sylib 218	1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ∅c0 4273  Oncon0 6323

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327

This theorem is referenced by:  snsn0non  6449  onxpdisj  6450  omabs  8587  cnfcom3lem  9624  0elold  27902  onexlimgt  43671  onexoegt  43672  oe0rif  43713  oege1  43734  onmcl  43759  omabs2  43760  omcl2  43761