Theorem snexxph 7016

Description: A case where the antecedent of snexg 4217 is not needed. The class { x | ph } is from dcextest 4617. (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 6481	. . 3 |- 1o e. On
2	1	elexi 2775	. 2 |- 1o e. _V
3		elsni 3640	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4165	. . . . . . . 8 |- -. _V e. _V
5		df-v 2765	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1715	. . . . . . . . . . . 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 2314	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2242	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2265	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 676	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1604	. . . . . . . . 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 2769	. . . . . . . 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 2766	. . . . . . . . . 10 |- y e. _V
19		biidd 172	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2908	. . . . . . . . 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 3495	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2229	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6498	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2287	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3187	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4171	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   _Vcvv 2763   (/)c0 3450   {csn 3622   Oncon0 4398   1oc1o 6467

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-1o 6474

This theorem is referenced by: (None)