Theorem elfi2 9370

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	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem elfi2

Step	Hyp	Ref	Expression
1		elex 3484	. . 3 ⊢ (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V))
3		simpr 489	. . . . 5 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝐴 = ∩ 𝑥)
4		eldifsni 4759	. . . . . . 7 ⊢ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) → 𝑥 ≠ ∅)
5	4	adantr 485	. . . . . 6 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝑥 ≠ ∅)
6		intex 5312	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
7	5, 6	sylib 221	. . . . 5 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → ∩ 𝑥 ∈ V)
8	3, 7	eqeltrd 2869	. . . 4 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝐴 ∈ V)
9	8	rexlimiva 3164	. . 3 ⊢ (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥 → 𝐴 ∈ V)
10	9	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥 → 𝐴 ∈ V))
11		elfi 9369	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
12		vprc 5282	. . . . . . . . . . 11 ⊢ ¬ V ∈ V
13		elsni 4608	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
14	13	inteqd 4918	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅} → ∩ 𝑥 = ∩ ∅)
15		int0 4928	. . . . . . . . . . . . 13 ⊢ ∩ ∅ = V
16	14, 15	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → ∩ 𝑥 = V)
17	16	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (∩ 𝑥 ∈ V ↔ V ∈ V))
18	12, 17	mtbiri 330	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → ¬ ∩ 𝑥 ∈ V)
19		simpr 489	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → 𝐴 = ∩ 𝑥)
20		simpll 778	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → 𝐴 ∈ V)
21	19, 20	eqeltrrd 2870	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → ∩ 𝑥 ∈ V)
22	18, 21	nsyl3 139	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → ¬ 𝑥 ∈ {∅})
23	22	biantrud 540	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅})))
24		eldif 3923	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅}))
25	23, 24	bitr4di 292	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
26	25	pm5.32da 589	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) ↔ (𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}))))
27		ancom 465	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ↔ (𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)))
28		ancom 465	. . . . . 6 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) ↔ (𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
29	26, 27, 28	3bitr4g 317	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ↔ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥)))
30	29	rexbidv2 3191	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥 ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
31	11, 30	bitrd 282	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
32	31	expcom 418	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥)))
33	2, 10, 32	pm5.21ndd 382	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912  ∅c0 4294  𝒫 cpw 4564  {csn 4591  ∩ cint 4913  ‘cfv 6534  Fincfn 8939  ficfi 9366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-fi 9367

This theorem is referenced by:  fifo  9388  firest  17481  alexsublem  24166  ispisys2  34484