Theorem disjss3 5069

Description: Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)

Assertion

Ref	Expression
disjss3	⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅) → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjss3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 3068	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅ ↔ ∀𝑥(𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅))
2		simprr 769	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
3		n0i 4264	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐶 → ¬ 𝐶 = ∅)
4	2, 3	syl 17	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → ¬ 𝐶 = ∅)
5		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 ∈ 𝐵)
6	5	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐵)
7		eldif 3893	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
8		simpl 482	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅))
9	7, 8	syl5bir 242	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → 𝐶 = ∅))
10	6, 9	mpand 691	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (¬ 𝑥 ∈ 𝐴 → 𝐶 = ∅))
11	4, 10	mt3d 148	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐴)
12	11, 2	jca 511	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
13	12	ex 412	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
14	13	alimi 1815	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ (𝐵 ∖ 𝐴) → 𝐶 = ∅) → ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
15	1, 14	sylbi 216	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅ → ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
16		moim 2544	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
17	15, 16	syl 17	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅ → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
18	17	alimdv 1920	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅ → (∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → ∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
19		dfdisj2 5037	. . . 4 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
20		dfdisj2 5037	. . . 4 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
21	18, 19, 20	3imtr4g 295	. . 3 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅ → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶))
22	21	adantl 481	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅) → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶))
23		disjss1 5041	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))
24	23	adantr 480	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅) → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))
25	22, 24	impbid 211	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅) → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  Disj wdisj 5035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rmo 3071  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-disj 5036

This theorem is referenced by:  carsggect  32185