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Theorem bj-snsetex 34173
Description: The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5182. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snsetex (𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-snsetex
Dummy variables 𝑦 𝑧 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elisset 3506 . . . 4 (𝐴𝑉 → ∃𝑦 𝑦 = 𝐴)
2 eleq2 2901 . . . . . . 7 (𝑦 = 𝐴 → ({𝑥} ∈ 𝑦 ↔ {𝑥} ∈ 𝐴))
32abbidv 2885 . . . . . 6 (𝑦 = 𝐴 → {𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴})
4 eleq1 2900 . . . . . . 7 ({𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴} → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V ↔ {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
54biimpd 230 . . . . . 6 ({𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴} → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
63, 5syl 17 . . . . 5 (𝑦 = 𝐴 → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
76eximi 1826 . . . 4 (∃𝑦 𝑦 = 𝐴 → ∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
81, 7syl 17 . . 3 (𝐴𝑉 → ∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
9 bj-eximcom 33874 . . . . 5 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
109com12 32 . . . 4 (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
11 ax-rep 5182 . . . . . . . 8 (∀𝑢𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) → ∃𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})))
12 19.3v 1977 . . . . . . . . . . 11 (∀𝑧 𝑢 = {𝑡} ↔ 𝑢 = {𝑡})
1312sbbii 2072 . . . . . . . . . . 11 ([𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
14 sbsbc 3775 . . . . . . . . . . . . . 14 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
15 sbceq2g 4367 . . . . . . . . . . . . . . 15 (𝑧 ∈ V → ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡}))
1615elv 3500 . . . . . . . . . . . . . 14 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡})
1714, 16bitri 276 . . . . . . . . . . . . 13 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡})
18 bj-csbsn 34119 . . . . . . . . . . . . . 14 𝑧 / 𝑡{𝑡} = {𝑧}
1918eqeq2i 2834 . . . . . . . . . . . . 13 (𝑢 = 𝑧 / 𝑡{𝑡} ↔ 𝑢 = {𝑧})
2017, 19bitri 276 . . . . . . . . . . . 12 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = {𝑧})
21 eqtr2 2842 . . . . . . . . . . . . 13 ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → {𝑡} = {𝑧})
22 vex 3498 . . . . . . . . . . . . . 14 𝑡 ∈ V
2322sneqr 4765 . . . . . . . . . . . . 13 ({𝑡} = {𝑧} → 𝑡 = 𝑧)
2421, 23syl 17 . . . . . . . . . . . 12 ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → 𝑡 = 𝑧)
2520, 24sylan2b 593 . . . . . . . . . . 11 ((𝑢 = {𝑡} ∧ [𝑧 / 𝑡]𝑢 = {𝑡}) → 𝑡 = 𝑧)
2612, 13, 25syl2anb 597 . . . . . . . . . 10 ((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
2726gen2 1788 . . . . . . . . 9 𝑡𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
28 nfa1 2146 . . . . . . . . . 10 𝑧𝑧 𝑢 = {𝑡}
2928mo 2645 . . . . . . . . 9 (∃𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) ↔ ∀𝑡𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧))
3027, 29mpbir 232 . . . . . . . 8 𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧)
3111, 30mpg 1789 . . . . . . 7 𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}))
32 bj-sbel1 34120 . . . . . . . . . . . 12 ([𝑡 / 𝑥]{𝑥} ∈ 𝑦𝑡 / 𝑥{𝑥} ∈ 𝑦)
33 bj-csbsn 34119 . . . . . . . . . . . . 13 𝑡 / 𝑥{𝑥} = {𝑡}
3433eleq1i 2903 . . . . . . . . . . . 12 (𝑡 / 𝑥{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
3532, 34bitri 276 . . . . . . . . . . 11 ([𝑡 / 𝑥]{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
36 df-clab 2800 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ [𝑡 / 𝑥]{𝑥} ∈ 𝑦)
3712anbi2i 622 . . . . . . . . . . . . . 14 ((𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ (𝑢𝑦𝑢 = {𝑡}))
38 eleq1a 2908 . . . . . . . . . . . . . . . . . 18 (𝑢𝑦 → ({𝑡} = 𝑢 → {𝑡} ∈ 𝑦))
3938com12 32 . . . . . . . . . . . . . . . . 17 ({𝑡} = 𝑢 → (𝑢𝑦 → {𝑡} ∈ 𝑦))
4039eqcoms 2829 . . . . . . . . . . . . . . . 16 (𝑢 = {𝑡} → (𝑢𝑦 → {𝑡} ∈ 𝑦))
4140imdistanri 570 . . . . . . . . . . . . . . 15 ((𝑢𝑦𝑢 = {𝑡}) → ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
42 eleq1a 2908 . . . . . . . . . . . . . . . 16 ({𝑡} ∈ 𝑦 → (𝑢 = {𝑡} → 𝑢𝑦))
4342impac 553 . . . . . . . . . . . . . . 15 (({𝑡} ∈ 𝑦𝑢 = {𝑡}) → (𝑢𝑦𝑢 = {𝑡}))
4441, 43impbii 210 . . . . . . . . . . . . . 14 ((𝑢𝑦𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
4537, 44bitri 276 . . . . . . . . . . . . 13 ((𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
4645exbii 1839 . . . . . . . . . . . 12 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}))
47 snex 5323 . . . . . . . . . . . . . 14 {𝑡} ∈ V
4847isseti 3509 . . . . . . . . . . . . 13 𝑢 𝑢 = {𝑡}
49 19.42v 1945 . . . . . . . . . . . . 13 (∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦 ∧ ∃𝑢 𝑢 = {𝑡}))
5048, 49mpbiran2 706 . . . . . . . . . . . 12 (∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
5146, 50bitri 276 . . . . . . . . . . 11 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
5235, 36, 513bitr4ri 305 . . . . . . . . . 10 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
5352bibi2i 339 . . . . . . . . 9 ((𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ (𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5453albii 1811 . . . . . . . 8 (∀𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∀𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5554exbii 1839 . . . . . . 7 (∃𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∃𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5631, 55mpbi 231 . . . . . 6 𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
57 dfcleq 2815 . . . . . . 7 (𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∀𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5857exbii 1839 . . . . . 6 (∃𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∃𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5956, 58mpbir 232 . . . . 5 𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦}
6059issetri 3511 . . . 4 {𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V
6110, 60mpg 1789 . . 3 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
628, 61syl 17 . 2 (𝐴𝑉 → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
63 ax5e 1904 . 2 (∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
6462, 63syl 17 1 (𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  [wsb 2060  wcel 2105  {cab 2799  Vcvv 3495  [wsbc 3771  csb 3882  {csn 4559
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4560  df-pr 4562
This theorem is referenced by:  bj-clex  34174  bj-snglex  34183
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