Theorem snexxph 6887

Description: A case where the antecedent of snexg 4144 is not needed. The class { x | ph } is from dcextest 4538. (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 6364	. . 3 |- 1o e. On
2	1	elexi 2724	. 2 |- 1o e. _V
3		elsni 3578	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4096	. . . . . . . 8 |- -. _V e. _V
5		df-v 2714	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1681	. . . . . . . . . . . 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 2275	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	syl5req 2203	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2226	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 665	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1570	. . . . . . . . 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 2718	. . . . . . . 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 616	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2715	. . . . . . . . . 10 |- y e. _V
19		biidd 171	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2856	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 658	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 132	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3438	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2190	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6381	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2248	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3132	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4102	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   {cab 2143   _Vcvv 2712   (/)c0 3394   {csn 3560   Oncon0 4322   1oc1o 6350

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4325  df-on 4327  df-suc 4330  df-1o 6357

This theorem is referenced by: (None)