Theorem iindif2m 3995

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 3553	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶)))
2		eldif 3175	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3	2	bicomi 132	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ 𝐶))
4	3	ralbii 2512	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
5		ralnex 2494	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6		eliun 3931	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	5, 6	xchbinxr 685	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 ↔ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
8	7	anbi2i 457	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
9	1, 4, 8	3bitr3g 222	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)))
10		vex 2775	. . . 4 ⊢ 𝑦 ∈ V
11		eliin 3932	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶)))
12	10, 11	ax-mp 5	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
13		eldif 3175	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
14	9, 12, 13	3bitr4g 223	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)))
15	14	eqrdv 2203	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  Vcvv 2772   ∖ cdif 3163  ∪ ciun 3927  ∩ ciin 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-iun 3929  df-iin 3930

This theorem is referenced by: (None)