Theorem snexxph 7233

Description: A case where the antecedent of snexg 4302 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4708. (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 6667	. . 3 ⊢ 1o ∈ On
2	1	elexi 2828	. 2 ⊢ 1o ∈ V
3		elsni 3712	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = {𝑥 ∣ 𝜑})
4		vprc 4247	. . . . . . . 8 ⊢ ¬ V ∈ V
5		df-v 2817	. . . . . . . . . 10 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1749	. . . . . . . . . . . 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 2354	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	eqtr2id 2280	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2303	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 682	. . . . . . 7 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13		19.8a 1639	. . . . . . . . 9 ⊢ (𝑦 = {𝑥 ∣ 𝜑} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
14	3, 13	syl 14	. . . . . . . 8 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
15		isset 2822	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
16	14, 15	sylibr 134	. . . . . . 7 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} ∈ V)
17	12, 16	nsyl3 631	. . . . . 6 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ¬ 𝜑)
18		vex 2818	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19		biidd 172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
20	18, 19	elab 2964	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
21	20	notbii 674	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
22	21	biimpri 133	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
23	22	eq0rdv 3557	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
24	17, 23	syl 14	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} = ∅)
25	3, 24	eqtrd 2267	. . . 4 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = ∅)
26		0lt1o 6686	. . . 4 ⊢ ∅ ∈ 1o
27	25, 26	eqeltrdi 2325	. . 3 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 ∈ 1o)
28	27	ssriv 3246	. 2 ⊢ {{𝑥 ∣ 𝜑}} ⊆ 1o
29	2, 28	ssexi 4253	1 ⊢ {{𝑥 ∣ 𝜑}} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  Vcvv 2815  ∅c0 3512  {csn 3694  Oncon0 4489  1_oc1o 6653

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660

This theorem is referenced by: (None)