Theorem bj-snsetex 38003

Description: The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5234. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-snsetex	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-snsetex

Dummy variables 𝑦 𝑧 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2845	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		eleq2 2852	. . . . 5 ⊢ (𝑦 = 𝐴 → ({𝑥} ∈ 𝑦 ↔ {𝑥} ∈ 𝐴))
3	2	abbidv 2829	. . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴})
4		eleq1 2851	. . . . 5 ⊢ ({𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴} → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V ↔ {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
5	4	biimpd 232	. . . 4 ⊢ ({𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴} → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
6	3, 5	syl 18	. . 3 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
7	6	eximi 1867	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 → ∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
8		bj-eximcom 37643	. . . 4 ⊢ (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
9	8	com12 33	. . 3 ⊢ (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
10		ax-rep 5234	. . . . . . 7 ⊢ (∀𝑢∃𝑧∀𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) → ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ ∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡})))
11		19.3v 2016	. . . . . . . . . 10 ⊢ (∀𝑧 𝑢 = {𝑡} ↔ 𝑢 = {𝑡})
12	11	sbbii 2114	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
13		sbsbc 3745	. . . . . . . . . . . . 13 ⊢ ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
14		sbceq2g 4379	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = ⦋𝑧 / 𝑡⦌{𝑡}))
15	14	elv 3458	. . . . . . . . . . . . 13 ⊢ ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = ⦋𝑧 / 𝑡⦌{𝑡})
16	13, 15	bitri 278	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = ⦋𝑧 / 𝑡⦌{𝑡})
17		bj-csbsn 37943	. . . . . . . . . . . . 13 ⊢ ⦋𝑧 / 𝑡⦌{𝑡} = {𝑧}
18	17	eqeq2i 2776	. . . . . . . . . . . 12 ⊢ (𝑢 = ⦋𝑧 / 𝑡⦌{𝑡} ↔ 𝑢 = {𝑧})
19	16, 18	bitri 278	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = {𝑧})
20		eqtr2 2784	. . . . . . . . . . . 12 ⊢ ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → {𝑡} = {𝑧})
21		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
22	21	sneqr 4797	. . . . . . . . . . . 12 ⊢ ({𝑡} = {𝑧} → 𝑡 = 𝑧)
23	20, 22	syl 18	. . . . . . . . . . 11 ⊢ ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → 𝑡 = 𝑧)
24	19, 23	sylan2b 606	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑡} ∧ [𝑧 / 𝑡]𝑢 = {𝑡}) → 𝑡 = 𝑧)
25	11, 12, 24	syl2anb 610	. . . . . . . . 9 ⊢ ((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
26	25	gen2 1828	. . . . . . . 8 ⊢ ∀𝑡∀𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
27		nfa1 2190	. . . . . . . . 9 ⊢ Ⅎ𝑧∀𝑧 𝑢 = {𝑡}
28	27	mo 2593	. . . . . . . 8 ⊢ (∃𝑧∀𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) ↔ ∀𝑡∀𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧))
29	26, 28	mpbir 234	. . . . . . 7 ⊢ ∃𝑧∀𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧)
30	10, 29	mpg 1829	. . . . . 6 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ ∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡}))
31		bj-sbel1 37944	. . . . . . . . . . 11 ⊢ ([𝑡 / 𝑥]{𝑥} ∈ 𝑦 ↔ ⦋𝑡 / 𝑥⦌{𝑥} ∈ 𝑦)
32		bj-csbsn 37943	. . . . . . . . . . . 12 ⊢ ⦋𝑡 / 𝑥⦌{𝑥} = {𝑡}
33	32	eleq1i 2854	. . . . . . . . . . 11 ⊢ (⦋𝑡 / 𝑥⦌{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
34	31, 33	bitri 278	. . . . . . . . . 10 ⊢ ([𝑡 / 𝑥]{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
35		df-clab 2742	. . . . . . . . . 10 ⊢ (𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ [𝑡 / 𝑥]{𝑥} ∈ 𝑦)
36	11	anbi2i 635	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ (𝑢 ∈ 𝑦 ∧ 𝑢 = {𝑡}))
37		eleq1a 2858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ 𝑦 → ({𝑡} = 𝑢 → {𝑡} ∈ 𝑦))
38	37	com12 33	. . . . . . . . . . . . . . . 16 ⊢ ({𝑡} = 𝑢 → (𝑢 ∈ 𝑦 → {𝑡} ∈ 𝑦))
39	38	eqcoms 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = {𝑡} → (𝑢 ∈ 𝑦 → {𝑡} ∈ 𝑦))
40	39	imdistanri 580	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑦 ∧ 𝑢 = {𝑡}) → ({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}))
41		eleq1a 2858	. . . . . . . . . . . . . . 15 ⊢ ({𝑡} ∈ 𝑦 → (𝑢 = {𝑡} → 𝑢 ∈ 𝑦))
42	41	impac 562	. . . . . . . . . . . . . 14 ⊢ (({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}) → (𝑢 ∈ 𝑦 ∧ 𝑢 = {𝑡}))
43	40, 42	impbii 212	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑦 ∧ 𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}))
44	36, 43	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}))
45	44	exbii 1880	. . . . . . . . . . 11 ⊢ (∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ∃𝑢({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}))
46		vsnex 5396	. . . . . . . . . . . . 13 ⊢ {𝑡} ∈ V
47	46	isseti 3471	. . . . . . . . . . . 12 ⊢ ∃𝑢 𝑢 = {𝑡}
48		19.42v 1987	. . . . . . . . . . . 12 ⊢ (∃𝑢({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦 ∧ ∃𝑢 𝑢 = {𝑡}))
49	47, 48	mpbiran2 723	. . . . . . . . . . 11 ⊢ (∃𝑢({𝑡} ∈ 𝑦 ∧ 𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
50	45, 49	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
51	34, 35, 50	3bitr4ri 307	. . . . . . . . 9 ⊢ (∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
52	51	bibi2i 340	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑧 ↔ ∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ (𝑡 ∈ 𝑧 ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
53	52	albii 1851	. . . . . . 7 ⊢ (∀𝑡(𝑡 ∈ 𝑧 ↔ ∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∀𝑡(𝑡 ∈ 𝑧 ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
54	53	exbii 1880	. . . . . 6 ⊢ (∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ ∃𝑢(𝑢 ∈ 𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
55	30, 54	mpbi 233	. . . . 5 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
56		dfcleq 2756	. . . . . 6 ⊢ (𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∀𝑡(𝑡 ∈ 𝑧 ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
57	56	exbii 1880	. . . . 5 ⊢ (∃𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
58	55, 57	mpbir 234	. . . 4 ⊢ ∃𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦}
59	58	issetri 3472	. . 3 ⊢ {𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V
60	9, 59	mpg 1829	. 2 ⊢ (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
61		ax5e 1946	. 2 ⊢ (∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
62	1, 7, 60, 61	4syl 20	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811  [wsb 2100   ∈ wcel 2146  {cab 2741  Vcvv 3453  [wsbc 3741  ⦋csb 3850  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-v 3455  df-sbc 3742  df-csb 3851  df-dif 4049  df-un 4153  df-nul 4281  df-sn 4588  df-pr 4590

This theorem is used by:  bj-clexab  38004  bj-snglex  38013