Theorem el1o 3115

Description: Membership in ordinal one.

Assertion

Ref	Expression
el1o	⊢ (A ∈ 1o ↔ A = ∅)

Proof of Theorem el1o

Step	Hyp	Ref	Expression
1		df1o2 3111	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 1153	. 2 ⊢ (A ∈ 1o ↔ A ∈ {∅})
3		0ex 1745	. . 3 ⊢ ∅ ∈ V
4	3	elsnc2 1832	. 2 ⊢ (A ∈ {∅} ↔ A = ∅)
5	2, 4	bitr 151	1 ⊢ (A ∈ 1o ↔ A = ∅)

Syntax hints:   ↔ wb 127   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {csn 1808  1_oc1o 3099

This theorem is referenced by:  0lt1o 3116

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-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812  df-suc 2205  df-1o 3104

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