Theorem snexxph 6838

Description: A case where the antecedent of snexg 4108 is not needed. The class { x | ph } is from dcextest 4495. (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 6320	. . 3 |- 1o e. On
2	1	elexi 2698	. 2 |- 1o e. _V
3		elsni 3545	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4060	. . . . . . . 8 |- -. _V e. _V
5		df-v 2688	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1677	. . . . . . . . . . . 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 2257	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	syl5req 2185	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2208	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 664	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1569	. . . . . . . . 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 2692	. . . . . . . 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 615	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2689	. . . . . . . . . 10 |- y e. _V
19		biidd 171	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2828	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 657	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 132	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3407	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2172	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6337	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2230	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3101	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4066	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   _Vcvv 2686   (/)c0 3363   {csn 3527   Oncon0 4285   1oc1o 6306

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-1o 6313

This theorem is referenced by: (None)