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 ∈ 2𝑜 → 𝐴 ⊆ 1𝑜)

Proof of Theorem el2oss1o

Step	Hyp	Ref	Expression
1		elpri 3469	. . 3 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2		df2o3 6195	. . 3 ⊢ 2𝑜 = {∅, 1𝑜}
3	1, 2	eleq2s 2182	. 2 ⊢ (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
4		0ss 3321	. . . 4 ⊢ ∅ ⊆ 1𝑜
5		sseq1 3047	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1𝑜 ↔ ∅ ⊆ 1𝑜))
6	4, 5	mpbiri 166	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1𝑜)
7		eqimss 3078	. . 3 ⊢ (𝐴 = 1𝑜 → 𝐴 ⊆ 1𝑜)
8	6, 7	jaoi 671	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → 𝐴 ⊆ 1𝑜)
9	3, 8	syl 14	1 ⊢ (𝐴 ∈ 2𝑜 → 𝐴 ⊆ 1𝑜)

Syntax hints:   → wi 4   ∨ wo 664   = wceq 1289   ∈ wcel 1438   ⊆ wss 2999  ∅c0 3286  {cpr 3447  1_𝑜c1o 6174  2_𝑜c2o 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