Theorem snexxph 6931

Description: A case where the antecedent of snexg 4171 is not needed. The class { x | ph } is from dcextest 4566. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)

Assertion

Ref	Expression
snexxph	|- { { x | ph } } e. _V

Distinct variable group:   ph, x

Proof of Theorem snexxph

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1on 6406	. . 3 |- 1o e. On
2	1	elexi 2743	. 2 |- 1o e. _V
3		elsni 3602	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4122	. . . . . . . 8 |- -. _V e. _V
5		df-v 2733	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1695	. . . . . . . . . . . 12 |- x = x
7		pm5.1im 172	. . . . . . . . . . . 12 |- ( x = x -> ( ph -> ( x = x <-> ph ) ) )
8	6, 7	ax-mp 5	. . . . . . . . . . 11 |- ( ph -> ( x = x <-> ph ) )
9	8	abbidv 2289	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2217	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2240	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 671	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1584	. . . . . . . . 9 |- ( y = { x | ph } -> E. y y = { x | ph } )
14	3, 13	syl 14	. . . . . . . 8 |- ( y e. { { x | ph } } -> E. y y = { x | ph } )
15		isset 2737	. . . . . . . 8 |- ( { x | ph } e. _V <-> E. y y = { x | ph } )
16	14, 15	sylibr 133	. . . . . . 7 |- ( y e. { { x | ph } } -> { x | ph } e. _V )
17	12, 16	nsyl3 622	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2734	. . . . . . . . . 10 |- y e. _V
19		biidd 171	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2875	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 664	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 132	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3460	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2204	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6423	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2262	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3152	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4128	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1349   E.wex 1486   e. wcel 2142   {cab 2157   _Vcvv 2731   (/)c0 3415   {csn 3584   Oncon0 4349   1oc1o 6392

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-pw 3569  df-sn 3590  df-pr 3591  df-uni 3798  df-tr 4089  df-iord 4352  df-on 4354  df-suc 4357  df-1o 6399

This theorem is referenced by: (None)