Theorem exss 3963

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 3245	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
2		df-rab 2315	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	eleq2i 2104	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4	3	exbii 1496	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
5	1, 4	bitr3i 175	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6		vex 2560	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	snss 3494	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8		ssab2 3024	. . . . . 6 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
9		sstr2 2952	. . . . . 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 114	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴)
12		simpr 103	. . . . . . . 8 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13		equsb1 1668	. . . . . . . . 9 ⊢ [𝑧 / 𝑥]𝑥 = 𝑧
14		velsn 3392	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
15	14	sbbii 1648	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
16	13, 15	mpbir 134	. . . . . . . 8 ⊢ [𝑧 / 𝑥]𝑥 ∈ {𝑧}
17	12, 16	jctil 295	. . . . . . 7 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18		df-clab 2027	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
19		sban 1829	. . . . . . . 8 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
20	18, 19	bitri 173	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
21		df-rab 2315	. . . . . . . . 9 ⊢ {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
22	21	eleq2i 2104	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23		df-clab 2027	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24		sban 1829	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
25	23, 24	bitri 173	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
26	22, 25	bitri 173	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
27	17, 20, 26	3imtr4i 190	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
28		elex2 2570	. . . . . 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 3245	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ∃𝑥 ∈ {𝑧}𝜑)
31	29, 30	sylib 127	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
32		snexgOLD 3935	. . . . . 6 ⊢ (𝑧 ∈ V → {𝑧} ∈ V)
33	6, 32	ax-mp 7	. . . . 5 ⊢ {𝑧} ∈ V
34		sseq1 2966	. . . . . 6 ⊢ (𝑦 = {𝑧} → (𝑦 ⊆ 𝐴 ↔ {𝑧} ⊆ 𝐴))
35		rexeq 2506	. . . . . 6 ⊢ (𝑦 = {𝑧} → (∃𝑥 ∈ 𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
36	34, 35	anbi12d 442	. . . . 5 ⊢ (𝑦 = {𝑧} → ((𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
37	33, 36	spcev 2647	. . . 4 ⊢ (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
38	11, 31, 37	syl2anc 391	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
39	38	exlimiv 1489	. 2 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
40	5, 39	sylbi 114	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  [wsb 1645  {cab 2026  ∃wrex 2307  {crab 2310  Vcvv 2557   ⊆ wss 2917  {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by: (None)