Theorem snexxph 7113

Description: A case where the antecedent of snexg 4267 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4672. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)

Assertion

Ref	Expression
snexxph	⊢ {{𝑥 ∣ 𝜑}} ∈ V

Distinct variable group:   𝜑,𝑥

Proof of Theorem snexxph

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1on 6567	. . 3 ⊢ 1o ∈ On
2	1	elexi 2812	. 2 ⊢ 1o ∈ V
3		elsni 3684	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = {𝑥 ∣ 𝜑})
4		vprc 4215	. . . . . . . 8 ⊢ ¬ V ∈ V
5		df-v 2801	. . . . . . . . . 10 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1747	. . . . . . . . . . . 12 ⊢ 𝑥 = 𝑥
7		pm5.1im 173	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥 ↔ 𝜑)))
8	6, 7	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = 𝑥 ↔ 𝜑))
9	8	abbidv 2347	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	eqtr2id 2275	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2298	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 679	. . . . . . 7 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13		19.8a 1636	. . . . . . . . 9 ⊢ (𝑦 = {𝑥 ∣ 𝜑} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
14	3, 13	syl 14	. . . . . . . 8 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
15		isset 2806	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
16	14, 15	sylibr 134	. . . . . . 7 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} ∈ V)
17	12, 16	nsyl3 629	. . . . . 6 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ¬ 𝜑)
18		vex 2802	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19		biidd 172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
20	18, 19	elab 2947	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
21	20	notbii 672	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
22	21	biimpri 133	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
23	22	eq0rdv 3536	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
24	17, 23	syl 14	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} = ∅)
25	3, 24	eqtrd 2262	. . . 4 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = ∅)
26		0lt1o 6584	. . . 4 ⊢ ∅ ∈ 1o
27	25, 26	eqeltrdi 2320	. . 3 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 ∈ 1o)
28	27	ssriv 3228	. 2 ⊢ {{𝑥 ∣ 𝜑}} ⊆ 1o
29	2, 28	ssexi 4221	1 ⊢ {{𝑥 ∣ 𝜑}} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  Vcvv 2799  ∅c0 3491  {csn 3666  Oncon0 4453  1_oc1o 6553

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-1o 6560

This theorem is referenced by: (None)