Theorem 0elsucexmid 4687

Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)

Hypothesis

Ref	Expression
0elsucexmid.1	|- A. x e. On (/) e. suc x

Assertion

Ref	Expression
0elsucexmid	|- ( ph \/ -. ph )

Distinct variable group:   ph, x

Proof of Theorem 0elsucexmid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtriexmidlem 4641	. . . 4 |- { y e. { (/) } | ph } e. On
2		0elsucexmid.1	. . . 4 |- A. x e. On (/) e. suc x
3		suceq 4523	. . . . . 6 |- ( x = { y e. { (/) } | ph } -> suc x = suc { y e. { (/) } | ph } )
4	3	eleq2d 2302	. . . . 5 |- ( x = { y e. { (/) } | ph } -> ( (/) e. suc x <-> (/) e. suc { y e. { (/) } | ph } ) )
5	4	rspcv 2917	. . . 4 |- ( { y e. { (/) } | ph } e. On -> ( A. x e. On (/) e. suc x -> (/) e. suc { y e. { (/) } | ph } ) )
6	1, 2, 5	mp2 16	. . 3 |- (/) e. suc { y e. { (/) } | ph }
7		0ex 4237	. . . 4 |- (/) e. _V
8	7	elsuc 4527	. . 3 |- ( (/) e. suc { y e. { (/) } | ph } <-> ( (/) e. { y e. { (/) } | ph } \/ (/) = { y e. { (/) } | ph } ) )
9	6, 8	mpbi 145	. 2 |- ( (/) e. { y e. { (/) } | ph } \/ (/) = { y e. { (/) } | ph } )
10	7	snid 3720	. . . . 5 |- (/) e. { (/) }
11		biidd 172	. . . . . 6 |- ( y = (/) -> ( ph <-> ph ) )
12	11	elrab3 2974	. . . . 5 |- ( (/) e. { (/) } -> ( (/) e. { y e. { (/) } | ph } <-> ph ) )
13	10, 12	ax-mp 5	. . . 4 |- ( (/) e. { y e. { (/) } | ph } <-> ph )
14	13	biimpi 120	. . 3 |- ( (/) e. { y e. { (/) } | ph } -> ph )
15		ordtriexmidlem2 4642	. . . 4 |- ( { y e. { (/) } | ph } = (/) -> -. ph )
16	15	eqcoms 2235	. . 3 |- ( (/) = { y e. { (/) } | ph } -> -. ph )
17	14, 16	orim12i 767	. 2 |- ( ( (/) e. { y e. { (/) } | ph } \/ (/) = { y e. { (/) } | ph } ) -> ( ph \/ -. ph ) )
18	9, 17	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   A.wral 2520   {crab 2524   (/)c0 3508   {csn 3689   Oncon0 4484   suc csuc 4486

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by: (None)