Theorem iundif2ss 3769

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 2993	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
2	1	rexbii 2379	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3		r19.42v 2517	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶))
4	2, 3	bitri 182	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶))
5		rexnalim 2364	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 → ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6		vex 2615	. . . . . . 7 ⊢ 𝑦 ∈ V
7		eliin 3709	. . . . . . 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 635	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 → ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
10	9	anim2i 334	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
11	4, 10	sylbi 119	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
12		eliun 3708	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
13		eldif 2993	. . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
14	11, 12, 13	3imtr4i 199	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) → 𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶))
15	14	ssriv 3014	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ⊆ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:  ¬ wn 3   ∧ wa 102   ↔ wb 103   ∈ wcel 1434  ∀wral 2353  ∃wrex 2354  Vcvv 2612   ∖ cdif 2981   ⊆ wss 2984  ∪ ciun 3704  ∩ ciin 3705

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997  df-iun 3706  df-iin 3707

This theorem is referenced by: (None)