Theorem el2oss1o 13359

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13360. (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 3555	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		df2o3 6335	. . 3 ⊢ 2o = {∅, 1o}
3	1, 2	eleq2s 2235	. 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
4		0ss 3406	. . . 4 ⊢ ∅ ⊆ 1o
5		sseq1 3125	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o))
6	4, 5	mpbiri 167	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o)
7		eqimss 3156	. . 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 1332   ∈ wcel 1481   ⊆ wss 3076  ∅c0 3368  {cpr 3533  1_oc1o 6314  2_oc2o 6315

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-suc 4301  df-1o 6321  df-2o 6322

This theorem is referenced by:  nninfsellemsuc  13383