Theorem ss1oel2o 14026

Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4184 which more directly illustrates the contrast with el2oss1o 6422. (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 4184	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2		df1o2 6408	. . . . 5 |- 1o = { (/) }
3	2	sseq2i 3174	. . . 4 |- ( x C_ 1o <-> x C_ { (/) } )
4		df2o2 6410	. . . . . 6 |- 2o = { (/) , { (/) } }
5	4	eleq2i 2237	. . . . 5 |- ( x e. 2o <-> x e. { (/) , { (/) } } )
6		vex 2733	. . . . . 6 |- x e. _V
7	6	elpr 3604	. . . . 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 1463	. 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 703   A.wal 1346   = wceq 1348   e. wcel 2141   C_ wss 3121   (/)c0 3414   {csn 3583   {cpr 3584  EXMIDwem 4180   1oc1o 6388   2oc2o 6389

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-exmid 4181  df-suc 4356  df-1o 6395  df-2o 6396

This theorem is referenced by: (None)