Theorem snexxph 7117

Description: A case where the antecedent of snexg 4268 is not needed. The class { x | ph } is from dcextest 4673. (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 6569	. . 3 |- 1o e. On
2	1	elexi 2812	. 2 |- 1o e. _V
3		elsni 3684	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4216	. . . . . . . 8 |- -. _V e. _V
5		df-v 2801	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1747	. . . . . . . . . . . 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 2347	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2275	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2298	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 679	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1636	. . . . . . . . 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 2806	. . . . . . . 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 629	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2802	. . . . . . . . . 10 |- y e. _V
19		biidd 172	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2947	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 672	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 133	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3536	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2262	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6586	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2320	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3228	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4222	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   _Vcvv 2799   (/)c0 3491   {csn 3666   Oncon0 4454   1oc1o 6555

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-1o 6562

This theorem is referenced by: (None)