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Theorem el2oss1o 11842
Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 11843. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
el2oss1o (𝐴 ∈ 2𝑜𝐴 ⊆ 1𝑜)

Proof of Theorem el2oss1o
StepHypRef Expression
1 elpri 3469 . . 3 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2 df2o3 6195 . . 3 2𝑜 = {∅, 1𝑜}
31, 2eleq2s 2182 . 2 (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
4 0ss 3321 . . . 4 ∅ ⊆ 1𝑜
5 sseq1 3047 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ 1𝑜 ↔ ∅ ⊆ 1𝑜))
64, 5mpbiri 166 . . 3 (𝐴 = ∅ → 𝐴 ⊆ 1𝑜)
7 eqimss 3078 . . 3 (𝐴 = 1𝑜𝐴 ⊆ 1𝑜)
86, 7jaoi 671 . 2 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → 𝐴 ⊆ 1𝑜)
93, 8syl 14 1 (𝐴 ∈ 2𝑜𝐴 ⊆ 1𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 664   = wceq 1289  wcel 1438  wss 2999  c0 3286  {cpr 3447  1𝑜c1o 6174  2𝑜c2o 6175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453  df-suc 4198  df-1o 6181  df-2o 6182
This theorem is referenced by:  nninfsellemsuc  11859
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