Theorem exss 4149

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 3390	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑)
2		df-rab 2425	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	eleq2i 2206	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4	3	exbii 1584	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
5	1, 4	bitr3i 185	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6		vex 2689	. . . . . 6 ⊢ 𝑧 ∈ V
7	6	snss 3649	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8		ssab2 3181	. . . . . 6 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
9		sstr2 3104	. . . . . 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 120	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴)
12		simpr 109	. . . . . . . 8 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13		equsb1 1758	. . . . . . . . 9 ⊢ [𝑧 / 𝑥]𝑥 = 𝑧
14		velsn 3544	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
15	14	sbbii 1738	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
16	13, 15	mpbir 145	. . . . . . . 8 ⊢ [𝑧 / 𝑥]𝑥 ∈ {𝑧}
17	12, 16	jctil 310	. . . . . . 7 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18		df-clab 2126	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
19		sban 1928	. . . . . . . 8 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
20	18, 19	bitri 183	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
21		df-rab 2425	. . . . . . . . 9 ⊢ {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
22	21	eleq2i 2206	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23		df-clab 2126	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24		sban 1928	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
25	23, 24	bitri 183	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
26	22, 25	bitri 183	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
27	17, 20, 26	3imtr4i 200	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
28		elex2 2702	. . . . . 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 3390	. . . . 5 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ∃𝑥 ∈ {𝑧}𝜑)
31	29, 30	sylib 121	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
32	6	snex 4109	. . . . 5 ⊢ {𝑧} ∈ V
33		sseq1 3120	. . . . . 6 ⊢ (𝑦 = {𝑧} → (𝑦 ⊆ 𝐴 ↔ {𝑧} ⊆ 𝐴))
34		rexeq 2627	. . . . . 6 ⊢ (𝑦 = {𝑧} → (∃𝑥 ∈ 𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
35	33, 34	anbi12d 464	. . . . 5 ⊢ (𝑦 = {𝑧} → ((𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
36	32, 35	spcev 2780	. . . 4 ⊢ (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
37	11, 31, 36	syl2anc 408	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
38	37	exlimiv 1577	. 2 ⊢ (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
39	5, 38	sylbi 120	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  [wsb 1735  {cab 2125  ∃wrex 2417  {crab 2420   ⊆ wss 3071  {csn 3527

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533

This theorem is referenced by: (None)