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Theorem exss 5346
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 (∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3144 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21neeq1i 3077 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅)
3 rabn0 4336 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
4 n0 4307 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
52, 3, 43bitr3i 302 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
6 vex 3495 . . . . . 6 𝑧 ∈ V
76snss 4710 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
8 ssab2 4052 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
9 sstr2 3971 . . . . . 6 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴 → {𝑧} ⊆ 𝐴))
108, 9mpi 20 . . . . 5 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
117, 10sylbi 218 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
12 simpr 485 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13 equsb1v 2103 . . . . . . . . 9 [𝑧 / 𝑥]𝑥 = 𝑧
14 velsn 4573 . . . . . . . . . 10 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
1514sbbii 2072 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
1613, 15mpbir 232 . . . . . . . 8 [𝑧 / 𝑥]𝑥 ∈ {𝑧}
1712, 16jctil 520 . . . . . . 7 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18 df-clab 2797 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ [𝑧 / 𝑥](𝑥𝐴𝜑))
19 sban 2077 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
2018, 19bitri 276 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
21 df-rab 3144 . . . . . . . . 9 {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
2221eleq2i 2901 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23 df-clab 2797 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24 sban 2077 . . . . . . . 8 ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2522, 23, 243bitri 298 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2617, 20, 253imtr4i 293 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
2726ne0d 4298 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅)
28 rabn0 4336 . . . . 5 ({𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ {𝑧}𝜑)
2927, 28sylib 219 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
30 snex 5322 . . . . 5 {𝑧} ∈ V
31 sseq1 3989 . . . . . 6 (𝑦 = {𝑧} → (𝑦𝐴 ↔ {𝑧} ⊆ 𝐴))
32 rexeq 3404 . . . . . 6 (𝑦 = {𝑧} → (∃𝑥𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
3331, 32anbi12d 630 . . . . 5 (𝑦 = {𝑧} → ((𝑦𝐴 ∧ ∃𝑥𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
3430, 33spcev 3604 . . . 4 (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3511, 29, 34syl2anc 584 . . 3 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3635exlimiv 1922 . 2 (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
375, 36sylbi 218 1 (∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  [wsb 2060  wcel 2105  {cab 2796  wne 3013  wrex 3136  {crab 3139  wss 3933  c0 4288  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by: (None)
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