Theorem el2oss1o 6552

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 16127. (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 3666	. . 3 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		df2o3 6539	. . 3 |- 2o = { (/) , 1o }
3	1, 2	eleq2s 2302	. 2 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
4		0ss 3507	. . . 4 |- (/) C_ 1o
5		sseq1 3224	. . . 4 |- ( A = (/) -> ( A C_ 1o <-> (/) C_ 1o ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = (/) -> A C_ 1o )
7		eqimss 3255	. . 3 |- ( A = 1o -> A C_ 1o )
8	6, 7	jaoi 718	. 2 |- ( ( A = (/) \/ A = 1o ) -> A C_ 1o )
9	3, 8	syl 14	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2178   C_ wss 3174   (/)c0 3468   {cpr 3644   1oc1o 6518   2oc2o 6519

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-pr 3650  df-suc 4436  df-1o 6525  df-2o 6526

This theorem is referenced by:  nnnninfeq2  7257  nninfwlpoimlemg  7303  nninfsellemsuc  16151