Theorem iindif2m 3819

Description: Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
iindif2m	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iindif2m

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28mv 3394	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶)))
2		eldif 3022	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3	2	bicomi 131	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ 𝐶))
4	3	ralbii 2395	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
5		ralnex 2380	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6		eliun 3756	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	5, 6	xchbinxr 646	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
8	7	anbi2i 446	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
9	1, 4, 8	3bitr3g 221	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)))
10		vex 2636	. . . 4 ⊢ 𝑦 ∈ V
11		eliin 3757	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶)))
12	10, 11	ax-mp 7	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
13		eldif 3022	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
14	9, 12, 13	3bitr4g 222	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)))
15	14	eqrdv 2093	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  Vcvv 2633   ∖ cdif 3010  ∪ ciun 3752  ∩ ciin 3753

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-iun 3754  df-iin 3755

This theorem is referenced by: (None)