Theorem ss1oel2o 11888

Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4032 which more directly illustrates the contrast with el2oss1o 11887. (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 4032	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2		df1o2 6194	. . . . 5 |- 1o = { (/) }
3	2	sseq2i 3051	. . . 4 |- ( x C_ 1o <-> x C_ { (/) } )
4		df2o2 6196	. . . . . 6 |- 2o = { (/) , { (/) } }
5	4	eleq2i 2154	. . . . 5 |- ( x e. 2o <-> x e. { (/) , { (/) } } )
6		vex 2622	. . . . . 6 |- x e. _V
7	6	elpr 3467	. . . . 5 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
8	5, 7	bitri 182	. . . 4 |- ( x e. 2o <-> ( x = (/) \/ x = { (/) } ) )
9	3, 8	imbi12i 237	. . 3 |- ( ( x C_ 1o -> x e. 2o ) <-> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
10	9	albii 1404	. 2 |- ( A. x ( x C_ 1o -> x e. 2o ) <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
11	1, 10	bitr4i 185	1 |- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 664   A.wal 1287   = wceq 1289   e. wcel 1438   C_ wss 2999   (/)c0 3286   {csn 3446   {cpr 3447  EXMIDwem 4029   1oc1o 6174   2oc2o 6175

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965

This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453  df-exmid 4030  df-suc 4198  df-1o 6181  df-2o 6182

This theorem is referenced by: (None)