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Theorem elfi2 8872
Description: The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
elfi2 (𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem elfi2
StepHypRef Expression
1 elex 3518 . . 3 (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V)
21a1i 11 . 2 (𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V))
3 simpr 485 . . . . 5 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝐴 = 𝑥)
4 eldifsni 4721 . . . . . . 7 (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) → 𝑥 ≠ ∅)
54adantr 481 . . . . . 6 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝑥 ≠ ∅)
6 intex 5237 . . . . . 6 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
75, 6sylib 219 . . . . 5 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝑥 ∈ V)
83, 7eqeltrd 2918 . . . 4 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝐴 ∈ V)
98rexlimiva 3286 . . 3 (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥𝐴 ∈ V)
109a1i 11 . 2 (𝐵𝑉 → (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥𝐴 ∈ V))
11 elfi 8871 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
12 vprc 5216 . . . . . . . . . . 11 ¬ V ∈ V
13 elsni 4581 . . . . . . . . . . . . . 14 (𝑥 ∈ {∅} → 𝑥 = ∅)
1413inteqd 4879 . . . . . . . . . . . . 13 (𝑥 ∈ {∅} → 𝑥 = ∅)
15 int0 4888 . . . . . . . . . . . . 13 ∅ = V
1614, 15syl6eq 2877 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = V)
1716eleq1d 2902 . . . . . . . . . . 11 (𝑥 ∈ {∅} → ( 𝑥 ∈ V ↔ V ∈ V))
1812, 17mtbiri 328 . . . . . . . . . 10 (𝑥 ∈ {∅} → ¬ 𝑥 ∈ V)
19 simpr 485 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝐴 = 𝑥)
20 simpll 763 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝐴 ∈ V)
2119, 20eqeltrrd 2919 . . . . . . . . . 10 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝑥 ∈ V)
2218, 21nsyl3 140 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → ¬ 𝑥 ∈ {∅})
2322biantrud 532 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅})))
24 eldif 3950 . . . . . . . 8 (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅}))
2523, 24syl6bbr 290 . . . . . . 7 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
2625pm5.32da 579 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((𝐴 = 𝑥𝑥 ∈ (𝒫 𝐵 ∩ Fin)) ↔ (𝐴 = 𝑥𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}))))
27 ancom 461 . . . . . 6 ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = 𝑥) ↔ (𝐴 = 𝑥𝑥 ∈ (𝒫 𝐵 ∩ Fin)))
28 ancom 461 . . . . . 6 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) ↔ (𝐴 = 𝑥𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
2926, 27, 283bitr4g 315 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = 𝑥) ↔ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥)))
3029rexbidv2 3300 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥 ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
3111, 30bitrd 280 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
3231expcom 414 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥)))
332, 10, 32pm5.21ndd 381 1 (𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wrex 3144  Vcvv 3500  cdif 3937  cin 3939  c0 4295  𝒫 cpw 4542  {csn 4564   cint 4874  cfv 6354  Fincfn 8503  ficfi 8868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-fi 8869
This theorem is referenced by:  fifo  8890  firest  16701  alexsublem  22587  ispisys2  31317
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