Theorem el2oss1o 6392

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13636. (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 3584	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		df2o3 6379	. . 3 ⊢ 2o = {∅, 1o}
3	1, 2	eleq2s 2252	. 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
4		0ss 3433	. . . 4 ⊢ ∅ ⊆ 1o
5		sseq1 3151	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o))
6	4, 5	mpbiri 167	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o)
7		eqimss 3182	. . 3 ⊢ (𝐴 = 1o → 𝐴 ⊆ 1o)
8	6, 7	jaoi 706	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → 𝐴 ⊆ 1o)
9	3, 8	syl 14	1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)

Syntax hints:   → wi 4   ∨ wo 698   = wceq 1335   ∈ wcel 2128   ⊆ wss 3102  ∅c0 3395  {cpr 3562  1_oc1o 6358  2_oc2o 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-sn 3567  df-pr 3568  df-suc 4333  df-1o 6365  df-2o 6366

This theorem is referenced by:  nnnninfeq2  7074  nninfsellemsuc  13655