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Theorem bj-snsetex 33799
Description: The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 5049. (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 3426 . . . 4 (𝐴𝑉 → ∃𝑦 𝑦 = 𝐴)
2 eleq2 2854 . . . . . . 7 (𝑦 = 𝐴 → ({𝑥} ∈ 𝑦 ↔ {𝑥} ∈ 𝐴))
32abbidv 2843 . . . . . 6 (𝑦 = 𝐴 → {𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴})
4 eleq1 2853 . . . . . . 7 ({𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴} → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V ↔ {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
54biimpd 221 . . . . . 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 1797 . . . 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 19.35 1840 . . . . . 6 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) ↔ (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
109biimpi 208 . . . . 5 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
1110com12 32 . . . 4 (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
12 ax-rep 5049 . . . . . . . 8 (∀𝑢𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) → ∃𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})))
13 19.3v 1939 . . . . . . . . . . 11 (∀𝑧 𝑢 = {𝑡} ↔ 𝑢 = {𝑡})
1413sbbii 2027 . . . . . . . . . . 11 ([𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
15 sbsbc 3685 . . . . . . . . . . . . . 14 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
16 sbceq2g 4254 . . . . . . . . . . . . . . 15 (𝑧 ∈ V → ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡}))
1716elv 3420 . . . . . . . . . . . . . 14 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡})
1815, 17bitri 267 . . . . . . . . . . . . 13 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡})
19 bj-csbsn 33719 . . . . . . . . . . . . . 14 𝑧 / 𝑡{𝑡} = {𝑧}
2019eqeq2i 2790 . . . . . . . . . . . . 13 (𝑢 = 𝑧 / 𝑡{𝑡} ↔ 𝑢 = {𝑧})
2118, 20bitri 267 . . . . . . . . . . . 12 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = {𝑧})
22 eqtr2 2800 . . . . . . . . . . . . 13 ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → {𝑡} = {𝑧})
23 vex 3418 . . . . . . . . . . . . . 14 𝑡 ∈ V
2423sneqr 4645 . . . . . . . . . . . . 13 ({𝑡} = {𝑧} → 𝑡 = 𝑧)
2522, 24syl 17 . . . . . . . . . . . 12 ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → 𝑡 = 𝑧)
2621, 25sylan2b 584 . . . . . . . . . . 11 ((𝑢 = {𝑡} ∧ [𝑧 / 𝑡]𝑢 = {𝑡}) → 𝑡 = 𝑧)
2713, 14, 26syl2anb 588 . . . . . . . . . 10 ((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
2827gen2 1759 . . . . . . . . 9 𝑡𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
29 nfa1 2088 . . . . . . . . . 10 𝑧𝑧 𝑢 = {𝑡}
3029mo 2579 . . . . . . . . 9 (∃𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) ↔ ∀𝑡𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧))
3128, 30mpbir 223 . . . . . . . 8 𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧)
3212, 31mpg 1760 . . . . . . 7 𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}))
33 bj-sbel1 33720 . . . . . . . . . . . 12 ([𝑡 / 𝑥]{𝑥} ∈ 𝑦𝑡 / 𝑥{𝑥} ∈ 𝑦)
34 bj-csbsn 33719 . . . . . . . . . . . . 13 𝑡 / 𝑥{𝑥} = {𝑡}
3534eleq1i 2856 . . . . . . . . . . . 12 (𝑡 / 𝑥{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
3633, 35bitri 267 . . . . . . . . . . 11 ([𝑡 / 𝑥]{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
37 df-clab 2759 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ [𝑡 / 𝑥]{𝑥} ∈ 𝑦)
3813anbi2i 613 . . . . . . . . . . . . . 14 ((𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ (𝑢𝑦𝑢 = {𝑡}))
39 eleq1a 2861 . . . . . . . . . . . . . . . . . 18 (𝑢𝑦 → ({𝑡} = 𝑢 → {𝑡} ∈ 𝑦))
4039com12 32 . . . . . . . . . . . . . . . . 17 ({𝑡} = 𝑢 → (𝑢𝑦 → {𝑡} ∈ 𝑦))
4140eqcoms 2786 . . . . . . . . . . . . . . . 16 (𝑢 = {𝑡} → (𝑢𝑦 → {𝑡} ∈ 𝑦))
4241imdistanri 562 . . . . . . . . . . . . . . 15 ((𝑢𝑦𝑢 = {𝑡}) → ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
43 eleq1a 2861 . . . . . . . . . . . . . . . 16 ({𝑡} ∈ 𝑦 → (𝑢 = {𝑡} → 𝑢𝑦))
4443impac 545 . . . . . . . . . . . . . . 15 (({𝑡} ∈ 𝑦𝑢 = {𝑡}) → (𝑢𝑦𝑢 = {𝑡}))
4542, 44impbii 201 . . . . . . . . . . . . . 14 ((𝑢𝑦𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
4638, 45bitri 267 . . . . . . . . . . . . 13 ((𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
4746exbii 1810 . . . . . . . . . . . 12 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}))
48 snex 5188 . . . . . . . . . . . . . 14 {𝑡} ∈ V
4948isseti 3429 . . . . . . . . . . . . 13 𝑢 𝑢 = {𝑡}
50 19.42v 1912 . . . . . . . . . . . . 13 (∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦 ∧ ∃𝑢 𝑢 = {𝑡}))
5149, 50mpbiran2 697 . . . . . . . . . . . 12 (∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
5247, 51bitri 267 . . . . . . . . . . 11 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
5336, 37, 523bitr4ri 296 . . . . . . . . . 10 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
5453bibi2i 330 . . . . . . . . 9 ((𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ (𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5554albii 1782 . . . . . . . 8 (∀𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∀𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5655exbii 1810 . . . . . . 7 (∃𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∃𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5732, 56mpbi 222 . . . . . 6 𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
58 dfcleq 2772 . . . . . . 7 (𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∀𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5958exbii 1810 . . . . . 6 (∃𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∃𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
6057, 59mpbir 223 . . . . 5 𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦}
6160issetri 3431 . . . 4 {𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V
6211, 61mpg 1760 . . 3 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
638, 62syl 17 . 2 (𝐴𝑉 → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
64 ax5e 1871 . 2 (∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
6563, 64syl 17 1 (𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505   = wceq 1507  wex 1742  [wsb 2015  wcel 2050  {cab 2758  Vcvv 3415  [wsbc 3681  csb 3786  {csn 4441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-nul 4179  df-sn 4442  df-pr 4444
This theorem is referenced by:  bj-clex  33800  bj-snglex  33809
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