Theorem disjen 8658

Description: A stronger form of pwuninel 7924. We can use pwuninel 7924, 2pwuninel 8656 to create one or two sets disjoint from a given set 𝐴, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set 𝐵 we can construct a set 𝑥 that is equinumerous to it and disjoint from 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.)

Assertion

Ref	Expression
disjen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))

Proof of Theorem disjen

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2nd2 7710	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
2	1	ad2antll 728	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
3		simprl 770	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 ∈ 𝐴)
4	2, 3	eqeltrrd 2891	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴)
5		fvex 6658	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
6		fvex 6658	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
7	5, 6	opelrn 5777	. . . . . 6 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 𝐴 → (2nd ‘𝑥) ∈ ran 𝐴)
8	4, 7	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ ran 𝐴)
9		pwuninel 7924	. . . . . 6 ⊢ ¬ 𝒫 ∪ ran 𝐴 ∈ ran 𝐴
10		xp2nd 7704	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴})
11	10	ad2antll 728	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴})
12		elsni 4542	. . . . . . . 8 ⊢ ((2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴} → (2nd ‘𝑥) = 𝒫 ∪ ran 𝐴)
13	11, 12	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) = 𝒫 ∪ ran 𝐴)
14	13	eleq1d 2874	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ((2nd ‘𝑥) ∈ ran 𝐴 ↔ 𝒫 ∪ ran 𝐴 ∈ ran 𝐴))
15	9, 14	mtbiri 330	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ¬ (2nd ‘𝑥) ∈ ran 𝐴)
16	8, 15	pm2.65da 816	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴})))
17		elin 3897	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴})))
18	16, 17	sylnibr 332	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})))
19	18	eq0rdv 4312	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅)
20		simpr 488	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
21		rnexg 7595	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
22	21	adantr 484	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V)
23		uniexg 7446	. . . 4 ⊢ (ran 𝐴 ∈ V → ∪ ran 𝐴 ∈ V)
24		pwexg 5244	. . . 4 ⊢ (∪ ran 𝐴 ∈ V → 𝒫 ∪ ran 𝐴 ∈ V)
25	22, 23, 24	3syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 ∪ ran 𝐴 ∈ V)
26		xpsneng 8585	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝒫 ∪ ran 𝐴 ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)
27	20, 25, 26	syl2anc 587	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)
28	19, 27	jca 515	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ⟨cop 4531  ∪ cuni 4800   class class class wbr 5030   × cxp 5517  ran crn 5520  ‘cfv 6324  1^st c1st 7669  2^nd c2nd 7670   ≈ cen 8489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1st 7671  df-2nd 7672  df-en 8493

This theorem is referenced by:  disjenex  8659  domss2  8660