Theorem snexxph 6846

Description: A case where the antecedent of snexg 4116 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4503. (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 6328	. . 3 ⊢ 1o ∈ On
2	1	elexi 2701	. 2 ⊢ 1o ∈ V
3		elsni 3550	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = {𝑥 ∣ 𝜑})
4		vprc 4068	. . . . . . . 8 ⊢ ¬ V ∈ V
5		df-v 2691	. . . . . . . . . 10 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
6		equid 1678	. . . . . . . . . . . 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 2258	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝜑})
10	5, 9	syl5req 2186	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ 𝜑} = V)
11	10	eleq1d 2209	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ 𝜑} ∈ V ↔ V ∈ V))
12	4, 11	mtbiri 665	. . . . . . 7 ⊢ (𝜑 → ¬ {𝑥 ∣ 𝜑} ∈ V)
13		19.8a 1570	. . . . . . . . 9 ⊢ (𝑦 = {𝑥 ∣ 𝜑} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
14	3, 13	syl 14	. . . . . . . 8 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
15		isset 2695	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑})
16	14, 15	sylibr 133	. . . . . . 7 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} ∈ V)
17	12, 16	nsyl3 616	. . . . . 6 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → ¬ 𝜑)
18		vex 2692	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19		biidd 171	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
20	18, 19	elab 2832	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
21	20	notbii 658	. . . . . . . 8 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ 𝜑)
22	21	biimpri 132	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
23	22	eq0rdv 3412	. . . . . 6 ⊢ (¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
24	17, 23	syl 14	. . . . 5 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → {𝑥 ∣ 𝜑} = ∅)
25	3, 24	eqtrd 2173	. . . 4 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 = ∅)
26		0lt1o 6345	. . . 4 ⊢ ∅ ∈ 1o
27	25, 26	eqeltrdi 2231	. . 3 ⊢ (𝑦 ∈ {{𝑥 ∣ 𝜑}} → 𝑦 ∈ 1o)
28	27	ssriv 3106	. 2 ⊢ {{𝑥 ∣ 𝜑}} ⊆ 1o
29	2, 28	ssexi 4074	1 ⊢ {{𝑥 ∣ 𝜑}} ∈ V

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  Vcvv 2689  ∅c0 3368  {csn 3532  Oncon0 4293  1_oc1o 6314

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301  df-1o 6321

This theorem is referenced by: (None)