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Theorem disjen 8662
Description: A stronger form of pwuninel 7930. We can use pwuninel 7930, 2pwuninel 8660 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 7717 . . . . . . . 8 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
21ad2antll 725 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3 simprl 767 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥𝐴)
42, 3eqeltrrd 2911 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
5 fvex 6676 . . . . . . 7 (1st𝑥) ∈ V
6 fvex 6676 . . . . . . 7 (2nd𝑥) ∈ V
75, 6opelrn 5806 . . . . . 6 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴 → (2nd𝑥) ∈ ran 𝐴)
84, 7syl 17 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ ran 𝐴)
9 pwuninel 7930 . . . . . 6 ¬ 𝒫 ran 𝐴 ∈ ran 𝐴
10 xp2nd 7711 . . . . . . . . 9 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
1110ad2antll 725 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
12 elsni 4574 . . . . . . . 8 ((2nd𝑥) ∈ {𝒫 ran 𝐴} → (2nd𝑥) = 𝒫 ran 𝐴)
1311, 12syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) = 𝒫 ran 𝐴)
1413eleq1d 2894 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ((2nd𝑥) ∈ ran 𝐴 ↔ 𝒫 ran 𝐴 ∈ ran 𝐴))
159, 14mtbiri 328 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ¬ (2nd𝑥) ∈ ran 𝐴)
168, 15pm2.65da 813 . . . 4 ((𝐴𝑉𝐵𝑊) → ¬ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
17 elin 4166 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
1816, 17sylnibr 330 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
1918eq0rdv 4354 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅)
20 simpr 485 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
21 rnexg 7603 . . . . 5 (𝐴𝑉 → ran 𝐴 ∈ V)
2221adantr 481 . . . 4 ((𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
23 uniexg 7456 . . . 4 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
24 pwexg 5270 . . . 4 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
2522, 23, 243syl 18 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
26 xpsneng 8590 . . 3 ((𝐵𝑊 ∧ 𝒫 ran 𝐴 ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2720, 25, 26syl2anc 584 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2819, 27jca 512 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  c0 4288  𝒫 cpw 4535  {csn 4557  cop 4563   cuni 4830   class class class wbr 5057   × cxp 5546  ran crn 5549  cfv 6348  1st c1st 7676  2nd c2nd 7677  cen 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-1st 7678  df-2nd 7679  df-en 8498
This theorem is referenced by:  disjenex  8663  domss2  8664
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