Theorem ss1oel2o 13360

Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4129 which more directly illustrates the contrast with el2oss1o 13359. (Contributed by Jim Kingdon, 8-Aug-2022.)

Assertion

Ref	Expression
ss1oel2o	|- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )

Proof of Theorem ss1oel2o

Step	Hyp	Ref	Expression
1		exmid01 4129	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2		df1o2 6334	. . . . 5 |- 1o = { (/) }
3	2	sseq2i 3129	. . . 4 |- ( x C_ 1o <-> x C_ { (/) } )
4		df2o2 6336	. . . . . 6 |- 2o = { (/) , { (/) } }
5	4	eleq2i 2207	. . . . 5 |- ( x e. 2o <-> x e. { (/) , { (/) } } )
6		vex 2692	. . . . . 6 |- x e. _V
7	6	elpr 3553	. . . . 5 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
8	5, 7	bitri 183	. . . 4 |- ( x e. 2o <-> ( x = (/) \/ x = { (/) } ) )
9	3, 8	imbi12i 238	. . 3 |- ( ( x C_ 1o -> x e. 2o ) <-> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
10	9	albii 1447	. 2 |- ( A. x ( x C_ 1o -> x e. 2o ) <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
11	1, 10	bitr4i 186	1 |- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   A.wal 1330   = wceq 1332   e. wcel 1481   C_ wss 3076   (/)c0 3368   {csn 3532   {cpr 3533  EXMIDwem 4126   1oc1o 6314   2oc2o 6315

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-exmid 4127  df-suc 4301  df-1o 6321  df-2o 6322

This theorem is referenced by: (None)