Theorem snexxph 7009

Description: A case where the antecedent of snexg 4213 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4613. (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 6476	. . 3 ⊢ 1o ∈ On
2	1	elexi 2772	. 2 ⊢ 1o ∈ V
3		elsni 3636	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = {𝑥 ∣ 𝜑})
4		vprc 4161	. . . . . . . 8 ⊢ ¬ V ∈ V
5		df-v 2762	. . . . . . . . . 10 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1712	. . . . . . . . . . . 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 2311	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	eqtr2id 2239	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2262	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 676	. . . . . . 7 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13		19.8a 1601	. . . . . . . . 9 ⊢ (𝑦 = {𝑥 ∣ 𝜑} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
14	3, 13	syl 14	. . . . . . . 8 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
15		isset 2766	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
16	14, 15	sylibr 134	. . . . . . 7 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} ∈ V)
17	12, 16	nsyl3 627	. . . . . 6 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ¬ 𝜑)
18		vex 2763	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19		biidd 172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
20	18, 19	elab 2904	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
21	20	notbii 669	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
22	21	biimpri 133	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
23	22	eq0rdv 3491	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
24	17, 23	syl 14	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} = ∅)
25	3, 24	eqtrd 2226	. . . 4 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = ∅)
26		0lt1o 6493	. . . 4 ⊢ ∅ ∈ 1o
27	25, 26	eqeltrdi 2284	. . 3 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 ∈ 1o)
28	27	ssriv 3183	. 2 ⊢ {{𝑥 ∣ 𝜑}} ⊆ 1o
29	2, 28	ssexi 4167	1 ⊢ {{𝑥 ∣ 𝜑}} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  Vcvv 2760  ∅c0 3446  {csn 3618  Oncon0 4394  1_oc1o 6462

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-1o 6469

This theorem is referenced by: (None)