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Theorem el2oss1o 6444
Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 14746. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
el2oss1o (𝐴 ∈ 2o𝐴 ⊆ 1o)

Proof of Theorem el2oss1o
StepHypRef Expression
1 elpri 3616 . . 3 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2 df2o3 6431 . . 3 2o = {∅, 1o}
31, 2eleq2s 2272 . 2 (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
4 0ss 3462 . . . 4 ∅ ⊆ 1o
5 sseq1 3179 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o))
64, 5mpbiri 168 . . 3 (𝐴 = ∅ → 𝐴 ⊆ 1o)
7 eqimss 3210 . . 3 (𝐴 = 1o𝐴 ⊆ 1o)
86, 7jaoi 716 . 2 ((𝐴 = ∅ ∨ 𝐴 = 1o) → 𝐴 ⊆ 1o)
93, 8syl 14 1 (𝐴 ∈ 2o𝐴 ⊆ 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708   = wceq 1353  wcel 2148  wss 3130  c0 3423  {cpr 3594  1oc1o 6410  2oc2o 6411
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-sn 3599  df-pr 3600  df-suc 4372  df-1o 6417  df-2o 6418
This theorem is referenced by:  nnnninfeq2  7127  nninfwlpoimlemg  7173  nninfsellemsuc  14764
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