Theorem elfi2 9437

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 3490	. . 3 ⊢ (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) → 𝐴 ∈ V))
3		simpr 484	. . . . 5 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝐴 = ∩ 𝑥)
4		eldifsni 4794	. . . . . . 7 ⊢ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) → 𝑥 ≠ ∅)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝑥 ≠ ∅)
6		intex 5339	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
7	5, 6	sylib 217	. . . . 5 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → ∩ 𝑥 ∈ V)
8	3, 7	eqeltrd 2829	. . . 4 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝐴 ∈ V)
9	8	rexlimiva 3144	. . 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 9436	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
12		vprc 5315	. . . . . . . . . . 11 ⊢ ¬ V ∈ V
13		elsni 4646	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
14	13	inteqd 4954	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅} → ∩ 𝑥 = ∩ ∅)
15		int0 4965	. . . . . . . . . . . . 13 ⊢ ∩ ∅ = V
16	14, 15	eqtrdi 2784	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → ∩ 𝑥 = V)
17	16	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → (∩ 𝑥 ∈ V ↔ V ∈ V))
18	12, 17	mtbiri 327	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → ¬ ∩ 𝑥 ∈ V)
19		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → 𝐴 = ∩ 𝑥)
20		simpll 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → 𝐴 ∈ V)
21	19, 20	eqeltrrd 2830	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → ∩ 𝑥 ∈ V)
22	18, 21	nsyl3 138	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → ¬ 𝑥 ∈ {∅})
23	22	biantrud 531	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅})))
24		eldif 3957	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ↔ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ ¬ 𝑥 ∈ {∅}))
25	23, 24	bitr4di 289	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 = ∩ 𝑥) → (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
26	25	pm5.32da 578	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) ↔ (𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}))))
27		ancom 460	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ↔ (𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)))
28		ancom 460	. . . . . 6 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) ↔ (𝐴 = ∩ 𝑥 ∧ 𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})))
29	26, 27, 28	3bitr4g 314	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ↔ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥)))
30	29	rexbidv2 3171	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥 ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
31	11, 30	bitrd 279	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
32	31	expcom 413	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥)))
33	2, 10, 32	pm5.21ndd 379	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  ∃wrex 3067  Vcvv 3471   ∖ cdif 3944   ∩ cin 3946  ∅c0 4323  𝒫 cpw 4603  {csn 4629  ∩ cint 4949  ‘cfv 6548  Fincfn 8963  ficfi 9433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-fi 9434

This theorem is referenced by:  fifo  9455  firest  17413  alexsublem  23947  ispisys2  33772