Theorem snexxph 6951

Description: A case where the antecedent of snexg 4186 is not needed. The class { x | ph } is from dcextest 4582. (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 6426	. . 3 |- 1o e. On
2	1	elexi 2751	. 2 |- 1o e. _V
3		elsni 3612	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4137	. . . . . . . 8 |- -. _V e. _V
5		df-v 2741	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1701	. . . . . . . . . . . 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 2295	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2223	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2246	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 675	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1590	. . . . . . . . 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 2745	. . . . . . . 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 626	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2742	. . . . . . . . . 10 |- y e. _V
19		biidd 172	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2883	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 668	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 133	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3469	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2210	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6443	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2268	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3161	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4143	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   _Vcvv 2739   (/)c0 3424   {csn 3594   Oncon0 4365   1oc1o 6412

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373  df-1o 6419

This theorem is referenced by: (None)