MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  exss Structured version   Visualization version   GIF version

Theorem exss 5121
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 3105 . . . 4 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21neeq1i 3042 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅)
3 rabn0 4158 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
4 n0 4132 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
52, 3, 43bitr3i 292 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
6 vex 3394 . . . . . 6 𝑧 ∈ V
76snss 4506 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
8 ssab2 3883 . . . . . 6 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
9 sstr2 3805 . . . . . 6 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴 → {𝑧} ⊆ 𝐴))
108, 9mpi 20 . . . . 5 ({𝑧} ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
117, 10sylbi 208 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑧} ⊆ 𝐴)
12 simpr 473 . . . . . . . 8 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑)
13 equsb1 2527 . . . . . . . . 9 [𝑧 / 𝑥]𝑥 = 𝑧
14 velsn 4386 . . . . . . . . . 10 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
1514sbbii 2067 . . . . . . . . 9 ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧)
1613, 15mpbir 222 . . . . . . . 8 [𝑧 / 𝑥]𝑥 ∈ {𝑧}
1712, 16jctil 511 . . . . . . 7 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
18 df-clab 2793 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ [𝑧 / 𝑥](𝑥𝐴𝜑))
19 sban 2558 . . . . . . . 8 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
2018, 19bitri 266 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
21 df-rab 3105 . . . . . . . . 9 {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}
2221eleq2i 2877 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)})
23 df-clab 2793 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑))
24 sban 2558 . . . . . . . . 9 ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2523, 24bitri 266 . . . . . . . 8 (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2622, 25bitri 266 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑))
2717, 20, 263imtr4i 283 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑})
2827ne0d 4123 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → {𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅)
29 rabn0 4158 . . . . 5 ({𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ {𝑧}𝜑)
3028, 29sylib 209 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑥 ∈ {𝑧}𝜑)
31 snex 5098 . . . . 5 {𝑧} ∈ V
32 sseq1 3823 . . . . . 6 (𝑦 = {𝑧} → (𝑦𝐴 ↔ {𝑧} ⊆ 𝐴))
33 rexeq 3328 . . . . . 6 (𝑦 = {𝑧} → (∃𝑥𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑))
3432, 33anbi12d 618 . . . . 5 (𝑦 = {𝑧} → ((𝑦𝐴 ∧ ∃𝑥𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑)))
3531, 34spcev 3493 . . . 4 (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3611, 30, 35syl2anc 575 . . 3 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
3736exlimiv 2021 . 2 (∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
385, 37sylbi 208 1 (∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wex 1859  [wsb 2060  wcel 2156  {cab 2792  wne 2978  wrex 3097  {crab 3100  wss 3769  c0 4116  {csn 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-sn 4371  df-pr 4373
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator