Theorem el2oss1o 13188

Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13189. (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 3550	. . 3 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		df2o3 6327	. . 3 |- 2o = { (/) , 1o }
3	1, 2	eleq2s 2234	. 2 |- ( A e. 2o -> ( A = (/) \/ A = 1o ) )
4		0ss 3401	. . . 4 |- (/) C_ 1o
5		sseq1 3120	. . . 4 |- ( A = (/) -> ( A C_ 1o <-> (/) C_ 1o ) )
6	4, 5	mpbiri 167	. . 3 |- ( A = (/) -> A C_ 1o )
7		eqimss 3151	. . 3 |- ( A = 1o -> A C_ 1o )
8	6, 7	jaoi 705	. 2 |- ( ( A = (/) \/ A = 1o ) -> A C_ 1o )
9	3, 8	syl 14	1 |- ( A e. 2o -> A C_ 1o )

Syntax hints:   -> wi 4   \/ wo 697   = wceq 1331   e. wcel 1480   C_ wss 3071   (/)c0 3363   {cpr 3528   1oc1o 6306   2oc2o 6307

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-suc 4293  df-1o 6313  df-2o 6314

This theorem is referenced by:  nninfsellemsuc  13208