Theorem iundif2ss 3993

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 3175	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
2	1	rexbii 2513	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3		r19.42v 2663	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶))
4	2, 3	bitri 184	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶))
5		rexnalim 2495	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 → ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6		vex 2775	. . . . . . 7 ⊢ 𝑦 ∈ V
7		eliin 3932	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
9	5, 8	sylnibr 679	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 → ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
10	9	anim2i 342	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
11	4, 10	sylbi 121	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
12		eliun 3931	. . 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	11, 12, 13	3imtr4i 201	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) → 𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶))
15	14	ssriv 3197	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ⊆ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  Vcvv 2772   ∖ cdif 3163   ⊆ wss 3166  ∪ 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-in 3172  df-ss 3179  df-iun 3929  df-iin 3930

This theorem is referenced by: (None)