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Theorem disjen 8061
Description: A stronger form of pwuninel 7346. We can use pwuninel 7346, 2pwuninel 8059 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 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))

Proof of Theorem disjen
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1st2nd2 7150 . . . . . . . 8 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
21ad2antll 764 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3 simprl 793 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥𝐴)
42, 3eqeltrrd 2699 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
5 fvex 6158 . . . . . . 7 (1st𝑥) ∈ V
6 fvex 6158 . . . . . . 7 (2nd𝑥) ∈ V
75, 6opelrn 5317 . . . . . 6 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴 → (2nd𝑥) ∈ ran 𝐴)
84, 7syl 17 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ ran 𝐴)
9 pwuninel 7346 . . . . . 6 ¬ 𝒫 ran 𝐴 ∈ ran 𝐴
10 xp2nd 7144 . . . . . . . . 9 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
1110ad2antll 764 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
12 elsni 4165 . . . . . . . 8 ((2nd𝑥) ∈ {𝒫 ran 𝐴} → (2nd𝑥) = 𝒫 ran 𝐴)
1311, 12syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) = 𝒫 ran 𝐴)
1413eleq1d 2683 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ((2nd𝑥) ∈ ran 𝐴 ↔ 𝒫 ran 𝐴 ∈ ran 𝐴))
159, 14mtbiri 317 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ¬ (2nd𝑥) ∈ ran 𝐴)
168, 15pm2.65da 599 . . . 4 ((𝐴𝑉𝐵𝑊) → ¬ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
17 elin 3774 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
1816, 17sylnibr 319 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
1918eq0rdv 3951 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅)
20 simpr 477 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
21 rnexg 7045 . . . . 5 (𝐴𝑉 → ran 𝐴 ∈ V)
2221adantr 481 . . . 4 ((𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
23 uniexg 6908 . . . 4 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
24 pwexg 4810 . . . 4 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
2522, 23, 243syl 18 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
26 xpsneng 7989 . . 3 ((𝐵𝑊 ∧ 𝒫 ran 𝐴 ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2720, 25, 26syl2anc 692 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2819, 27jca 554 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  c0 3891  𝒫 cpw 4130  {csn 4148  cop 4154   cuni 4402   class class class wbr 4613   × cxp 5072  ran crn 5075  cfv 5847  1st c1st 7111  2nd c2nd 7112  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-en 7900
This theorem is referenced by:  disjenex  8062  domss2  8063
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