Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-snsetex Structured version   Visualization version   GIF version

Theorem bj-snsetex 33282
 Description: The class of sets "whose singletons" belong to a set is a set. Nice application of ax-rep 4904. (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 3367 . . . 4 (𝐴𝑉 → ∃𝑦 𝑦 = 𝐴)
2 eleq2 2839 . . . . . . 7 (𝑦 = 𝐴 → ({𝑥} ∈ 𝑦 ↔ {𝑥} ∈ 𝐴))
32abbidv 2890 . . . . . 6 (𝑦 = 𝐴 → {𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴})
4 eleq1 2838 . . . . . . 7 ({𝑥 ∣ {𝑥} ∈ 𝑦} = {𝑥 ∣ {𝑥} ∈ 𝐴} → ({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V ↔ {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
54biimpd 219 . . . . . 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 1910 . . . 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 1957 . . . . . 6 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) ↔ (∀𝑦{𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V))
109biimpi 206 . . . . 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 4904 . . . . . . . 8 (∀𝑢𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) → ∃𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})))
13 19.3v 2066 . . . . . . . . . . 11 (∀𝑧 𝑢 = {𝑡} ↔ 𝑢 = {𝑡})
1413sbbii 2056 . . . . . . . . . . 11 ([𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
15 sbsbc 3591 . . . . . . . . . . . . . 14 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ [𝑧 / 𝑡]𝑢 = {𝑡})
16 vex 3354 . . . . . . . . . . . . . . 15 𝑧 ∈ V
17 sbceq2g 4134 . . . . . . . . . . . . . . 15 (𝑧 ∈ V → ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡}))
1816, 17ax-mp 5 . . . . . . . . . . . . . 14 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡})
1915, 18bitri 264 . . . . . . . . . . . . 13 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = 𝑧 / 𝑡{𝑡})
20 bj-csbsn 33228 . . . . . . . . . . . . . 14 𝑧 / 𝑡{𝑡} = {𝑧}
2120eqeq2i 2783 . . . . . . . . . . . . 13 (𝑢 = 𝑧 / 𝑡{𝑡} ↔ 𝑢 = {𝑧})
2219, 21bitri 264 . . . . . . . . . . . 12 ([𝑧 / 𝑡]𝑢 = {𝑡} ↔ 𝑢 = {𝑧})
23 eqtr2 2791 . . . . . . . . . . . . 13 ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → {𝑡} = {𝑧})
24 vex 3354 . . . . . . . . . . . . . 14 𝑡 ∈ V
2524sneqr 4504 . . . . . . . . . . . . 13 ({𝑡} = {𝑧} → 𝑡 = 𝑧)
2623, 25syl 17 . . . . . . . . . . . 12 ((𝑢 = {𝑡} ∧ 𝑢 = {𝑧}) → 𝑡 = 𝑧)
2722, 26sylan2b 581 . . . . . . . . . . 11 ((𝑢 = {𝑡} ∧ [𝑧 / 𝑡]𝑢 = {𝑡}) → 𝑡 = 𝑧)
2813, 14, 27syl2anb 585 . . . . . . . . . 10 ((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
2928gen2 1871 . . . . . . . . 9 𝑡𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧)
30 nfa1 2184 . . . . . . . . . 10 𝑧𝑧 𝑢 = {𝑡}
3130mo 2657 . . . . . . . . 9 (∃𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧) ↔ ∀𝑡𝑧((∀𝑧 𝑢 = {𝑡} ∧ [𝑧 / 𝑡]∀𝑧 𝑢 = {𝑡}) → 𝑡 = 𝑧))
3229, 31mpbir 221 . . . . . . . 8 𝑧𝑡(∀𝑧 𝑢 = {𝑡} → 𝑡 = 𝑧)
3312, 32mpg 1872 . . . . . . 7 𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}))
34 bj-sbel1 33229 . . . . . . . . . . . 12 ([𝑡 / 𝑥]{𝑥} ∈ 𝑦𝑡 / 𝑥{𝑥} ∈ 𝑦)
35 bj-csbsn 33228 . . . . . . . . . . . . 13 𝑡 / 𝑥{𝑥} = {𝑡}
3635eleq1i 2841 . . . . . . . . . . . 12 (𝑡 / 𝑥{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
3734, 36bitri 264 . . . . . . . . . . 11 ([𝑡 / 𝑥]{𝑥} ∈ 𝑦 ↔ {𝑡} ∈ 𝑦)
38 df-clab 2758 . . . . . . . . . . 11 (𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ [𝑡 / 𝑥]{𝑥} ∈ 𝑦)
3913anbi2i 609 . . . . . . . . . . . . . 14 ((𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ (𝑢𝑦𝑢 = {𝑡}))
40 eleq1a 2845 . . . . . . . . . . . . . . . . . 18 (𝑢𝑦 → ({𝑡} = 𝑢 → {𝑡} ∈ 𝑦))
4140com12 32 . . . . . . . . . . . . . . . . 17 ({𝑡} = 𝑢 → (𝑢𝑦 → {𝑡} ∈ 𝑦))
4241eqcoms 2779 . . . . . . . . . . . . . . . 16 (𝑢 = {𝑡} → (𝑢𝑦 → {𝑡} ∈ 𝑦))
4342imdistanri 559 . . . . . . . . . . . . . . 15 ((𝑢𝑦𝑢 = {𝑡}) → ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
44 eleq1a 2845 . . . . . . . . . . . . . . . 16 ({𝑡} ∈ 𝑦 → (𝑢 = {𝑡} → 𝑢𝑦))
4544impac 542 . . . . . . . . . . . . . . 15 (({𝑡} ∈ 𝑦𝑢 = {𝑡}) → (𝑢𝑦𝑢 = {𝑡}))
4643, 45impbii 199 . . . . . . . . . . . . . 14 ((𝑢𝑦𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
4739, 46bitri 264 . . . . . . . . . . . . 13 ((𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦𝑢 = {𝑡}))
4847exbii 1924 . . . . . . . . . . . 12 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ ∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}))
49 snex 5036 . . . . . . . . . . . . . 14 {𝑡} ∈ V
5049isseti 3361 . . . . . . . . . . . . 13 𝑢 𝑢 = {𝑡}
51 19.42v 2033 . . . . . . . . . . . . 13 (∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}) ↔ ({𝑡} ∈ 𝑦 ∧ ∃𝑢 𝑢 = {𝑡}))
5250, 51mpbiran2 689 . . . . . . . . . . . 12 (∃𝑢({𝑡} ∈ 𝑦𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
5348, 52bitri 264 . . . . . . . . . . 11 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ {𝑡} ∈ 𝑦)
5437, 38, 533bitr4ri 293 . . . . . . . . . 10 (∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡}) ↔ 𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
5554bibi2i 326 . . . . . . . . 9 ((𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ (𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5655albii 1895 . . . . . . . 8 (∀𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∀𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5756exbii 1924 . . . . . . 7 (∃𝑧𝑡(𝑡𝑧 ↔ ∃𝑢(𝑢𝑦 ∧ ∀𝑧 𝑢 = {𝑡})) ↔ ∃𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
5833, 57mpbi 220 . . . . . 6 𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦})
59 dfcleq 2765 . . . . . . 7 (𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∀𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
6059exbii 1924 . . . . . 6 (∃𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦} ↔ ∃𝑧𝑡(𝑡𝑧𝑡 ∈ {𝑥 ∣ {𝑥} ∈ 𝑦}))
6158, 60mpbir 221 . . . . 5 𝑧 𝑧 = {𝑥 ∣ {𝑥} ∈ 𝑦}
6261issetri 3362 . . . 4 {𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V
6311, 62mpg 1872 . . 3 (∃𝑦({𝑥 ∣ {𝑥} ∈ 𝑦} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V) → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
648, 63syl 17 . 2 (𝐴𝑉 → ∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
65 ax5e 1993 . 2 (∃𝑦{𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
6664, 65syl 17 1 (𝐴𝑉 → {𝑥 ∣ {𝑥} ∈ 𝐴} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   = wceq 1631  ∃wex 1852  [wsb 2049   ∈ wcel 2145  {cab 2757  Vcvv 3351  [wsbc 3587  ⦋csb 3682  {csn 4316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4317  df-pr 4319 This theorem is referenced by:  bj-clex  33283  bj-snglex  33292
 Copyright terms: Public domain W3C validator