Theorem el2oss1o 6529

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 15928. (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 3656	. . 3 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		df2o3 6516	. . 3 |- 2o = { (/) , 1o }
3	1, 2	eleq2s 2300	. 2 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
4		0ss 3499	. . . 4 |- (/) C_ 1o
5		sseq1 3216	. . . 4 |- ( A = (/) -> ( A C_ 1o <-> (/) C_ 1o ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = (/) -> A C_ 1o )
7		eqimss 3247	. . 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 2176   C_ wss 3166   (/)c0 3460   {cpr 3634   1oc1o 6495   2oc2o 6496

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-suc 4418  df-1o 6502  df-2o 6503

This theorem is referenced by:  nnnninfeq2  7231  nninfwlpoimlemg  7277  nninfsellemsuc  15949