Theorem el2oss1o 6457

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 15015. (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 3627	. . 3 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		df2o3 6444	. . 3 |- 2o = { (/) , 1o }
3	1, 2	eleq2s 2282	. 2 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
4		0ss 3473	. . . 4 |- (/) C_ 1o
5		sseq1 3190	. . . 4 |- ( A = (/) -> ( A C_ 1o <-> (/) C_ 1o ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = (/) -> A C_ 1o )
7		eqimss 3221	. . 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 1363   e. wcel 2158   C_ wss 3141   (/)c0 3434   {cpr 3605   1oc1o 6423   2oc2o 6424

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-suc 4383  df-1o 6430  df-2o 6431

This theorem is referenced by:  nnnninfeq2  7140  nninfwlpoimlemg  7186  nninfsellemsuc  15033