Theorem exss 4312

Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)

Assertion

Ref	Expression
exss	⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exss

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabn0m 3519	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
2		df-rab 2517	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	eleq2i 2296	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4	3	exbii 1651	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
5	1, 4	bitr3i 186	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6		vex 2802	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	snss 3802	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8		ssab2 3308	. . . . . 6 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
9		sstr2 3231	. . . . . 6 ⊢ ({𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 → {𝑧} ⊆ 𝐴))
10	8, 9	mpi 15	. . . . 5 ⊢ ({𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴)
11	7, 10	sylbi 121	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴)
12		simpr 110	. . . . . . . 8 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13		equsb1 1831	. . . . . . . . 9 ⊢ [𝑧 / 𝑥]𝑥 = 𝑧
14		velsn 3683	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
15	14	sbbii 1811	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
16	13, 15	mpbir 146	. . . . . . . 8 ⊢ [𝑧 / 𝑥]𝑥 ∈ {𝑧}
17	12, 16	jctil 312	. . . . . . 7 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18		df-clab 2216	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
19		sban 2006	. . . . . . . 8 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
20	18, 19	bitri 184	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
21		df-rab 2517	. . . . . . . . 9 ⊢ {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
22	21	eleq2i 2296	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23		df-clab 2216	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24		sban 2006	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
25	23, 24	bitri 184	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
26	22, 25	bitri 184	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
27	17, 20, 26	3imtr4i 201	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
28		elex2 2816	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} → ∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
29	27, 28	syl 14	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
30		rabn0m 3519	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ∃𝑥 ∈ {𝑧}𝜑)
31	29, 30	sylib 122	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
32	6	snex 4268	. . . . 5 ⊢ {𝑧} ∈ V
33		sseq1 3247	. . . . . 6 ⊢ (𝑦 = {𝑧} → (𝑦 ⊆ 𝐴 ↔ {𝑧} ⊆ 𝐴))
34		rexeq 2729	. . . . . 6 ⊢ (𝑦 = {𝑧} → (∃𝑥 ∈ 𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
35	33, 34	anbi12d 473	. . . . 5 ⊢ (𝑦 = {𝑧} → ((𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
36	32, 35	spcev 2898	. . . 4 ⊢ (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
37	11, 31, 36	syl2anc 411	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
38	37	exlimiv 1644	. 2 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
39	5, 38	sylbi 121	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538  [wsb 1808   ∈ wcel 2200  {cab 2215  ∃wrex 2509  {crab 2512   ⊆ wss 3197  {csn 3666

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672

This theorem is referenced by: (None)