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Theorem elfi2 8485
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 3352 . . 3 (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V)
21a1i 11 . 2 (𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V))
3 simpr 479 . . . . 5 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝐴 = 𝑥)
4 eldifsni 4466 . . . . . . 7 (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) → 𝑥 ≠ ∅)
54adantr 472 . . . . . 6 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝑥 ≠ ∅)
6 intex 4969 . . . . . 6 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
75, 6sylib 208 . . . . 5 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝑥 ∈ V)
83, 7eqeltrd 2839 . . . 4 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝐴 ∈ V)
98rexlimiva 3166 . . 3 (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥𝐴 ∈ V)
109a1i 11 . 2 (𝐵𝑉 → (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥𝐴 ∈ V))
11 elfi 8484 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
12 vprc 4948 . . . . . . . . . . 11 ¬ V ∈ V
13 elsni 4338 . . . . . . . . . . . . . 14 (𝑥 ∈ {∅} → 𝑥 = ∅)
1413inteqd 4632 . . . . . . . . . . . . 13 (𝑥 ∈ {∅} → 𝑥 = ∅)
15 int0 4642 . . . . . . . . . . . . 13 ∅ = V
1614, 15syl6eq 2810 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = V)
1716eleq1d 2824 . . . . . . . . . . 11 (𝑥 ∈ {∅} → ( 𝑥 ∈ V ↔ V ∈ V))
1812, 17mtbiri 316 . . . . . . . . . 10 (𝑥 ∈ {∅} → ¬ 𝑥 ∈ V)
19 simpr 479 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝐴 = 𝑥)
20 simpll 807 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝐴 ∈ V)
2119, 20eqeltrrd 2840 . . . . . . . . . 10 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝑥 ∈ V)
2218, 21nsyl3 133 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → ¬ 𝑥 ∈ {∅})
2322biantrud 529 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅})))
24 eldif 3725 . . . . . . . 8 (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅}))
2523, 24syl6bbr 278 . . . . . . 7 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
2625pm5.32da 676 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((𝐴 = 𝑥𝑥 ∈ (𝒫 𝐵 ∩ Fin)) ↔ (𝐴 = 𝑥𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}))))
27 ancom 465 . . . . . 6 ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = 𝑥) ↔ (𝐴 = 𝑥𝑥 ∈ (𝒫 𝐵 ∩ Fin)))
28 ancom 465 . . . . . 6 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) ↔ (𝐴 = 𝑥𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
2926, 27, 283bitr4g 303 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = 𝑥) ↔ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥)))
3029rexbidv2 3186 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥 ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
3111, 30bitrd 268 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
3231expcom 450 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥)))
332, 10, 32pm5.21ndd 368 1 (𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  c0 4058  𝒫 cpw 4302  {csn 4321   cint 4627  cfv 6049  Fincfn 8121  ficfi 8481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-fi 8482
This theorem is referenced by:  fifo  8503  firest  16295  alexsublem  22049  ispisys2  30525
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