Theorem on0eqelt 2370

Description: An ordinal number either equals zero or contains zero.

Assertion

Ref	Expression
on0eqelt	⊢ (A ∈ On → (A = ∅ ∨ ∅ ∈ A))

Proof of Theorem on0eqelt

Step	Hyp	Ref	Expression
1		0ss 1725	. . 3 ⊢ ∅ ⊆ A
2		0elon 2277	. . . 4 ⊢ ∅ ∈ On
3		onsseleq 2254	. . . 4 ⊢ ((∅ ∈ On ∧ A ∈ On) → (∅ ⊆ A ↔ (∅ ∈ A ∨ ∅ = A)))
4	2, 3	mpan 518	. . 3 ⊢ (A ∈ On → (∅ ⊆ A ↔ (∅ ∈ A ∨ ∅ = A)))
5	1, 4	mpbii 168	. 2 ⊢ (A ∈ On → (∅ ∈ A ∨ ∅ = A))
6		cleqcom 1103	. . . 4 ⊢ (∅ = A ↔ A = ∅)
7	6	orbi2i 214	. . 3 ⊢ ((∅ ∈ A ∨ ∅ = A) ↔ (∅ ∈ A ∨ A = ∅))
8		orcom 209	. . 3 ⊢ ((∅ ∈ A ∨ A = ∅) ↔ (A = ∅ ∨ ∅ ∈ A))
9	7, 8	bitr 151	. 2 ⊢ ((∅ ∈ A ∨ ∅ = A) ↔ (A = ∅ ∨ ∅ ∈ A))
10	5, 9	sylib 173	1 ⊢ (A ∈ On → (A = ∅ ∨ ∅ ∈ A))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∅c0 1707  Oncon0 2199

This theorem is referenced by:  onxpdisj 2476

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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