Theorem elfi2 9431

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 3485	. . 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 4771	. . . . . . 7 ⊢ (𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) → 𝑥 ≠ ∅)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝑥 ≠ ∅)
6		intex 5319	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
7	5, 6	sylib 218	. . . . 5 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → ∩ 𝑥 ∈ V)
8	3, 7	eqeltrd 2835	. . . 4 ⊢ ((𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅}) ∧ 𝐴 = ∩ 𝑥) → 𝐴 ∈ V)
9	8	rexlimiva 3134	. . 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 9430	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
12		vprc 5290	. . . . . . . . . . 11 ⊢ ¬ V ∈ V
13		elsni 4623	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
14	13	inteqd 4932	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅} → ∩ 𝑥 = ∩ ∅)
15		int0 4943	. . . . . . . . . . . . 13 ⊢ ∩ ∅ = V
16	14, 15	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {∅} → ∩ 𝑥 = V)
17	16	eleq1d 2820	. . . . . . . . . . 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 2836	. . . . . . . . . 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 3941	. . . . . . . 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 579	. . . . . 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 3161	. . . 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ∩ cin 3930  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ∩ cint 4927  ‘cfv 6536  Fincfn 8964  ficfi 9427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-fi 9428

This theorem is referenced by:  fifo  9449  firest  17451  alexsublem  23987  ispisys2  34189