Theorem el2oss1o 6502

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 15648. (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 3646	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		df2o3 6489	. . 3 ⊢ 2o = {∅, 1o}
3	1, 2	eleq2s 2291	. 2 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
4		0ss 3490	. . . 4 ⊢ ∅ ⊆ 1o
5		sseq1 3207	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ 1o ↔ ∅ ⊆ 1o))
6	4, 5	mpbiri 168	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ 1o)
7		eqimss 3238	. . 3 ⊢ (𝐴 = 1o → 𝐴 ⊆ 1o)
8	6, 7	jaoi 717	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → 𝐴 ⊆ 1o)
9	3, 8	syl 14	1 ⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)

Syntax hints:   → wi 4   ∨ wo 709   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157  ∅c0 3451  {cpr 3624  1_oc1o 6468  2_oc2o 6469

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-pr 3630  df-suc 4407  df-1o 6475  df-2o 6476

This theorem is referenced by:  nnnninfeq2  7196  nninfwlpoimlemg  7242  nninfsellemsuc  15666