Theorem snexxph 7257

Description: A case where the antecedent of snexg 4316 is not needed. The class { x | ph } is from dcextest 4723. (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 6684	. . 3 |- 1o e. On
2	1	elexi 2834	. 2 |- 1o e. _V
3		elsni 3723	. . . . 5 |- ( y e. { { x | ph } } -> y = { x | ph } )
4		vprc 4260	. . . . . . . 8 |- -. _V e. _V
5		df-v 2823	. . . . . . . . . 10 |- _V = { x | x = x }
6		equid 1753	. . . . . . . . . . . 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 2358	. . . . . . . . . 10 |- ( ph -> { x | x = x } = { x | ph } )
10	5, 9	eqtr2id 2284	. . . . . . . . 9 |- ( ph -> { x | ph } = _V )
11	10	eleq1d 2307	. . . . . . . 8 |- ( ph -> ( { x | ph } e. _V <-> _V e. _V ) )
12	4, 11	mtbiri 686	. . . . . . 7 |- ( ph -> -. { x | ph } e. _V )
13		19.8a 1643	. . . . . . . . 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 2828	. . . . . . . 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 635	. . . . . 6 |- ( y e. { { x | ph } } -> -. ph )
18		vex 2824	. . . . . . . . . 10 |- y e. _V
19		biidd 172	. . . . . . . . . 10 |- ( x = y -> ( ph <-> ph ) )
20	18, 19	elab 2970	. . . . . . . . 9 |- ( y e. { x | ph } <-> ph )
21	20	notbii 678	. . . . . . . 8 |- ( -. y e. { x | ph } <-> -. ph )
22	21	biimpri 133	. . . . . . 7 |- ( -. ph -> -. y e. { x | ph } )
23	22	eq0rdv 3570	. . . . . 6 |- ( -. ph -> { x | ph } = (/) )
24	17, 23	syl 14	. . . . 5 |- ( y e. { { x | ph } } -> { x | ph } = (/) )
25	3, 24	eqtrd 2271	. . . 4 |- ( y e. { { x | ph } } -> y = (/) )
26		0lt1o 6703	. . . 4 |- (/) e. 1o
27	25, 26	eqeltrdi 2329	. . 3 |- ( y e. { { x | ph } } -> y e. 1o )
28	27	ssriv 3252	. 2 |- { { x | ph } } C_ 1o
29	2, 28	ssexi 4266	1 |- { { x | ph } } e. _V

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   _Vcvv 2821   (/)c0 3520   {csn 3705   Oncon0 4503   1oc1o 6670

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677

This theorem is referenced by: (None)