Theorem snexxph 7148

Description: A case where the antecedent of snexg 4274 is not needed. The class { x | ph } is from dcextest 4679. (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 6588	. . 3 |- 1o e. On
2	1	elexi 2815	. 2 |- 1o e. _V
3		elsni 3687	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4221	. . . . . . . 8 |- -. _V e. _V
5		df-v 2804	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1749	. . . . . . . . . . . 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 2349	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2277	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2300	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 681	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1638	. . . . . . . . 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 2809	. . . . . . . 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 631	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2805	. . . . . . . . . 10 |- y e. _V
19		biidd 172	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2950	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 674	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 133	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3539	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2264	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6607	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2322	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3231	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4227	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   _Vcvv 2802   (/)c0 3494   {csn 3669   Oncon0 4460   1oc1o 6574

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468  df-1o 6581

This theorem is referenced by: (None)