Theorem ss1oel2o 15489

Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4227 which more directly illustrates the contrast with el2oss1o 6496. (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 4227	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2		df1o2 6482	. . . . 5 |- 1o = { (/) }
3	2	sseq2i 3206	. . . 4 |- ( x C_ 1o <-> x C_ { (/) } )
4		df2o2 6484	. . . . . 6 |- 2o = { (/) , { (/) } }
5	4	eleq2i 2260	. . . . 5 |- ( x e. 2o <-> x e. { (/) , { (/) } } )
6		vex 2763	. . . . . 6 |- x e. _V
7	6	elpr 3639	. . . . 5 |- ( x e. { (/) , { (/) } } <-> ( x = (/) \/ x = { (/) } ) )
8	5, 7	bitri 184	. . . 4 |- ( x e. 2o <-> ( x = (/) \/ x = { (/) } ) )
9	3, 8	imbi12i 239	. . 3 |- ( ( x C_ 1o -> x e. 2o ) <-> ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
10	9	albii 1481	. 2 |- ( A. x ( x C_ 1o -> x e. 2o ) <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
11	1, 10	bitr4i 187	1 |- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   e. wcel 2164   C_ wss 3153   (/)c0 3446   {csn 3618   {cpr 3619  EXMIDwem 4223   1oc1o 6462   2oc2o 6463

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-exmid 4224  df-suc 4402  df-1o 6469  df-2o 6470

This theorem is referenced by: (None)