Theorem snexxph 6927

Description: A case where the antecedent of snexg 4170 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4565. (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 6402	. . 3 ⊢ 1o ∈ On
2	1	elexi 2742	. 2 ⊢ 1o ∈ V
3		elsni 3601	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = {𝑥 ∣ 𝜑})
4		vprc 4121	. . . . . . . 8 ⊢ ¬ V ∈ V
5		df-v 2732	. . . . . . . . . 10 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1694	. . . . . . . . . . . 12 ⊢ 𝑥 = 𝑥
7		pm5.1im 172	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥 ↔ 𝜑)))
8	6, 7	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = 𝑥 ↔ 𝜑))
9	8	abbidv 2288	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	eqtr2id 2216	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2239	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 670	. . . . . . 7 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13		19.8a 1583	. . . . . . . . 9 ⊢ (𝑦 = {𝑥 ∣ 𝜑} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
14	3, 13	syl 14	. . . . . . . 8 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
15		isset 2736	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
16	14, 15	sylibr 133	. . . . . . 7 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} ∈ V)
17	12, 16	nsyl3 621	. . . . . 6 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ¬ 𝜑)
18		vex 2733	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19		biidd 171	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
20	18, 19	elab 2874	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
21	20	notbii 663	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
22	21	biimpri 132	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
23	22	eq0rdv 3459	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
24	17, 23	syl 14	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} = ∅)
25	3, 24	eqtrd 2203	. . . 4 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = ∅)
26		0lt1o 6419	. . . 4 ⊢ ∅ ∈ 1o
27	25, 26	eqeltrdi 2261	. . 3 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 ∈ 1o)
28	27	ssriv 3151	. 2 ⊢ {{𝑥 ∣ 𝜑}} ⊆ 1o
29	2, 28	ssexi 4127	1 ⊢ {{𝑥 ∣ 𝜑}} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  Vcvv 2730  ∅c0 3414  {csn 3583  Oncon0 4348  1_oc1o 6388

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-1o 6395

This theorem is referenced by: (None)