Theorem ss1oel2o 15605

Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4231 which more directly illustrates the contrast with el2oss1o 6501. (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 4231	. 2 |- (EXMID <-> A. x ( x C_ { (/) } -> ( x = (/) \/ x = { (/) } ) ) )
2		df1o2 6487	. . . . 5 |- 1o = { (/) }
3	2	sseq2i 3210	. . . 4 |- ( x C_ 1o <-> x C_ { (/) } )
4		df2o2 6489	. . . . . 6 |- 2o = { (/) , { (/) } }
5	4	eleq2i 2263	. . . . 5 |- ( x e. 2o <-> x e. { (/) , { (/) } } )
6		vex 2766	. . . . . 6 |- x e. _V
7	6	elpr 3643	. . . . 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 1484	. 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 2167   C_ wss 3157   (/)c0 3450   {csn 3622   {cpr 3623  EXMIDwem 4227   1oc1o 6467   2oc2o 6468

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-pr 3629  df-exmid 4228  df-suc 4406  df-1o 6474  df-2o 6475

This theorem is referenced by: (None)