Theorem el2oss1o 6411

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13873. (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 3599	. . 3 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		df2o3 6398	. . 3 |- 2o = { (/) , 1o }
3	1, 2	eleq2s 2261	. 2 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
4		0ss 3447	. . . 4 |- (/) C_ 1o
5		sseq1 3165	. . . 4 |- ( A = (/) -> ( A C_ 1o <-> (/) C_ 1o ) )
6	4, 5	mpbiri 167	. . 3 |- ( A = (/) -> A C_ 1o )
7		eqimss 3196	. . 3 |- ( A = 1o -> A C_ 1o )
8	6, 7	jaoi 706	. 2 |- ( ( A = (/) \/ A = 1o ) -> A C_ 1o )
9	3, 8	syl 14	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1343   e. wcel 2136   C_ wss 3116   (/)c0 3409   {cpr 3577   1oc1o 6377   2oc2o 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-suc 4349  df-1o 6384  df-2o 6385

This theorem is referenced by:  nnnninfeq2  7093  nninfsellemsuc  13892