Theorem el2oss1o 6542

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 16066. (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 3661	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		df2o3 6529	. . 3 ⊢ 2o = {∅, 1o}
3	1, 2	eleq2s 2301	. 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
4		0ss 3503	. . . 4 ⊢ ∅ ⊆ 1o
5		sseq1 3220	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o))
6	4, 5	mpbiri 168	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o)
7		eqimss 3251	. . 3 ⊢ (𝐴 = 1o → 𝐴 ⊆ 1o)
8	6, 7	jaoi 718	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → 𝐴 ⊆ 1o)
9	3, 8	syl 14	1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)

Syntax hints:   → wi 4   ∨ wo 710   = wceq 1373   ∈ wcel 2177   ⊆ wss 3170  ∅c0 3464  {cpr 3639  1_oc1o 6508  2_oc2o 6509

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-sn 3644  df-pr 3645  df-suc 4426  df-1o 6515  df-2o 6516

This theorem is referenced by:  nnnninfeq2  7246  nninfwlpoimlemg  7292  nninfsellemsuc  16090