Theorem el1o 6700

Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)

Assertion

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

Proof of Theorem el1o

Step	Hyp	Ref	Expression
1		df1o2 6691	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2305	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 4255	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 3739	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 184	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∅c0 3520  {csn 3705  1_oc1o 6670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3711  df-suc 4511  df-1o 6677

This theorem is referenced by:  0lt1o  6703  map0e  6957  map1  7091  1dom1el  7097  omp1eomlem  7424  ctmlemr  7438  ctssdclemn0  7440  exmidfodomrlemeldju  7541  exmidfodomrlemreseldju  7542  pw1on  7575  1tonninf  10856