Theorem iundif2ss 3713

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

Assertion

Ref	Expression
iundif2ss	⊢ ∪ x ∈ A (B ∖ 𝐶) ⊆ (B ∖ ∩ x ∈ A 𝐶)

Distinct variable group:   x,B

Allowed substitution hints:   A(x)   𝐶(x)

Proof of Theorem iundif2ss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 2921	. . . . . 6 ⊢ (y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ 𝐶))
2	1	rexbii 2325	. . . . 5 ⊢ (∃x ∈ A y ∈ (B ∖ 𝐶) ↔ ∃x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶))
3		r19.42v 2461	. . . . 5 ⊢ (∃x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶) ↔ (y ∈ B ∧ ∃x ∈ A ¬ y ∈ 𝐶))
4	2, 3	bitri 173	. . . 4 ⊢ (∃x ∈ A y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ∃x ∈ A ¬ y ∈ 𝐶))
5		rexnalim 2311	. . . . . 6 ⊢ (∃x ∈ A ¬ y ∈ 𝐶 → ¬ ∀x ∈ A y ∈ 𝐶)
6		vex 2554	. . . . . . 7 ⊢ y ∈ V
7		eliin 3653	. . . . . . 7 ⊢ (y ∈ V → (y ∈ ∩ x ∈ A 𝐶 ↔ ∀x ∈ A y ∈ 𝐶))
8	6, 7	ax-mp 7	. . . . . 6 ⊢ (y ∈ ∩ x ∈ A 𝐶 ↔ ∀x ∈ A y ∈ 𝐶)
9	5, 8	sylnibr 601	. . . . 5 ⊢ (∃x ∈ A ¬ y ∈ 𝐶 → ¬ y ∈ ∩ x ∈ A 𝐶)
10	9	anim2i 324	. . . 4 ⊢ ((y ∈ B ∧ ∃x ∈ A ¬ y ∈ 𝐶) → (y ∈ B ∧ ¬ y ∈ ∩ x ∈ A 𝐶))
11	4, 10	sylbi 114	. . 3 ⊢ (∃x ∈ A y ∈ (B ∖ 𝐶) → (y ∈ B ∧ ¬ y ∈ ∩ x ∈ A 𝐶))
12		eliun 3652	. . 3 ⊢ (y ∈ ∪ x ∈ A (B ∖ 𝐶) ↔ ∃x ∈ A y ∈ (B ∖ 𝐶))
13		eldif 2921	. . 3 ⊢ (y ∈ (B ∖ ∩ x ∈ A 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∩ x ∈ A 𝐶))
14	11, 12, 13	3imtr4i 190	. 2 ⊢ (y ∈ ∪ x ∈ A (B ∖ 𝐶) → y ∈ (B ∖ ∩ x ∈ A 𝐶))
15	14	ssriv 2943	1 ⊢ ∪ x ∈ A (B ∖ 𝐶) ⊆ (B ∖ ∩ x ∈ A 𝐶)

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ∖ cdif 2908   ⊆ wss 2911  ∪ ciun 3648  ∩ ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-iun 3650  df-iin 3651

This theorem is referenced by: (None)