Theorem snexxph 7009

Description: A case where the antecedent of snexg 4213 is not needed. The class { x | ph } is from dcextest 4613. (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 6476	. . 3 |- 1o e. On
2	1	elexi 2772	. 2 |- 1o e. _V
3		elsni 3636	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4161	. . . . . . . 8 |- -. _V e. _V
5		df-v 2762	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1712	. . . . . . . . . . . 12 |- x = x
7		pm5.1im 173	. . . . . . . . . . . 12 |- ( x = x -> ( ph -> ( x = x <-> ph ) ) )
8	6, 7	ax-mp 5	. . . . . . . . . . 11 |- ( ph -> ( x = x <-> ph ) )
9	8	abbidv 2311	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2239	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2262	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 676	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1601	. . . . . . . . 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 2766	. . . . . . . 8 |- ( { x | ph } e. _V <-> E. y y = { x | ph } )
16	14, 15	sylibr 134	. . . . . . 7 |- ( y e. { { x | ph } } -> { x | ph } e. _V )
17	12, 16	nsyl3 627	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2763	. . . . . . . . . 10 |- y e. _V
19		biidd 172	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2904	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 669	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 133	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3491	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2226	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6493	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2284	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3183	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4167	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   _Vcvv 2760   (/)c0 3446   {csn 3618   Oncon0 4394   1oc1o 6462

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-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-1o 6469

This theorem is referenced by: (None)