Theorem el2oss1o 6501

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 15605. (Contributed by Jim Kingdon, 8-Aug-2022.)

Assertion

Ref	Expression
el2oss1o	|- ( A e. 2o -> A C_ 1o )

Proof of Theorem el2oss1o

Step	Hyp	Ref	Expression
1		elpri 3645	. . 3 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		df2o3 6488	. . 3 |- 2o = { (/) , 1o }
3	1, 2	eleq2s 2291	. 2 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
4		0ss 3489	. . . 4 |- (/) C_ 1o
5		sseq1 3206	. . . 4 |- ( A = (/) -> ( A C_ 1o <-> (/) C_ 1o ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = (/) -> A C_ 1o )
7		eqimss 3237	. . 3 |- ( A = 1o -> A C_ 1o )
8	6, 7	jaoi 717	. 2 |- ( ( A = (/) \/ A = 1o ) -> A C_ 1o )
9	3, 8	syl 14	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2167   C_ wss 3157   (/)c0 3450   {cpr 3623   1oc1o 6467   2oc2o 6468

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 3451  df-sn 3628  df-pr 3629  df-suc 4406  df-1o 6474  df-2o 6475

This theorem is referenced by:  nnnninfeq2  7193  nninfwlpoimlemg  7239  nninfsellemsuc  15623