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Theorem elfi2 8476
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 3364 . . 3 (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V)
21a1i 11 . 2 (𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V))
3 simpr 471 . . . . 5 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝐴 = 𝑥)
4 eldifsni 4457 . . . . . . 7 (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) → 𝑥 ≠ ∅)
54adantr 466 . . . . . 6 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝑥 ≠ ∅)
6 intex 4951 . . . . . 6 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
75, 6sylib 208 . . . . 5 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝑥 ∈ V)
83, 7eqeltrd 2850 . . . 4 ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = 𝑥) → 𝐴 ∈ V)
98rexlimiva 3176 . . 3 (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥𝐴 ∈ V)
109a1i 11 . 2 (𝐵𝑉 → (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥𝐴 ∈ V))
11 elfi 8475 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
12 vprc 4931 . . . . . . . . . . 11 ¬ V ∈ V
13 elsni 4333 . . . . . . . . . . . . . 14 (𝑥 ∈ {∅} → 𝑥 = ∅)
1413inteqd 4616 . . . . . . . . . . . . 13 (𝑥 ∈ {∅} → 𝑥 = ∅)
15 int0 4625 . . . . . . . . . . . . 13 ∅ = V
1614, 15syl6eq 2821 . . . . . . . . . . . 12 (𝑥 ∈ {∅} → 𝑥 = V)
1716eleq1d 2835 . . . . . . . . . . 11 (𝑥 ∈ {∅} → ( 𝑥 ∈ V ↔ V ∈ V))
1812, 17mtbiri 316 . . . . . . . . . 10 (𝑥 ∈ {∅} → ¬ 𝑥 ∈ V)
19 simpr 471 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝐴 = 𝑥)
20 simpll 750 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝐴 ∈ V)
2119, 20eqeltrrd 2851 . . . . . . . . . 10 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → 𝑥 ∈ V)
2218, 21nsyl3 135 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → ¬ 𝑥 ∈ {∅})
2322biantrud 521 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ 𝐴 = 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅})))
24 eldif 3733 . . . . . . . 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 568 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝑉) → ((𝐴 = 𝑥𝑥 ∈ (𝒫 𝐵 ∩ Fin)) ↔ (𝐴 = 𝑥𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}))))
27 ancom 452 . . . . . 6 ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = 𝑥) ↔ (𝐴 = 𝑥𝑥 ∈ (𝒫 𝐵 ∩ Fin)))
28 ancom 452 . . . . . 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 3196 . . . 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 398 . 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 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  cdif 3720  cin 3722  c0 4063  𝒫 cpw 4297  {csn 4316   cint 4611  cfv 6031  Fincfn 8109  ficfi 8472
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  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-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-fi 8473
This theorem is referenced by:  fifo  8494  firest  16301  alexsublem  22068  ispisys2  30556
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