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Theorem disjen 8284
 Description: A stronger form of pwuninel 7571. We can use pwuninel 7571, 2pwuninel 8282 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 7373 . . . . . . . 8 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
21ad2antll 767 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3 simprl 811 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → 𝑥𝐴)
42, 3eqeltrrd 2840 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
5 fvex 6363 . . . . . . 7 (1st𝑥) ∈ V
6 fvex 6363 . . . . . . 7 (2nd𝑥) ∈ V
75, 6opelrn 5512 . . . . . 6 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴 → (2nd𝑥) ∈ ran 𝐴)
84, 7syl 17 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ ran 𝐴)
9 pwuninel 7571 . . . . . 6 ¬ 𝒫 ran 𝐴 ∈ ran 𝐴
10 xp2nd 7367 . . . . . . . . 9 (𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
1110ad2antll 767 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) ∈ {𝒫 ran 𝐴})
12 elsni 4338 . . . . . . . 8 ((2nd𝑥) ∈ {𝒫 ran 𝐴} → (2nd𝑥) = 𝒫 ran 𝐴)
1311, 12syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → (2nd𝑥) = 𝒫 ran 𝐴)
1413eleq1d 2824 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ((2nd𝑥) ∈ ran 𝐴 ↔ 𝒫 ran 𝐴 ∈ ran 𝐴))
159, 14mtbiri 316 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴}))) → ¬ (2nd𝑥) ∈ ran 𝐴)
168, 15pm2.65da 601 . . . 4 ((𝐴𝑉𝐵𝑊) → ¬ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
17 elin 3939 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × {𝒫 ran 𝐴})))
1816, 17sylnibr 318 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
1918eq0rdv 4122 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅)
20 simpr 479 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
21 rnexg 7264 . . . . 5 (𝐴𝑉 → ran 𝐴 ∈ V)
2221adantr 472 . . . 4 ((𝐴𝑉𝐵𝑊) → ran 𝐴 ∈ V)
23 uniexg 7121 . . . 4 (ran 𝐴 ∈ V → ran 𝐴 ∈ V)
24 pwexg 4999 . . . 4 ( ran 𝐴 ∈ V → 𝒫 ran 𝐴 ∈ V)
2522, 23, 243syl 18 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 ran 𝐴 ∈ V)
26 xpsneng 8212 . . 3 ((𝐵𝑊 ∧ 𝒫 ran 𝐴 ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2720, 25, 26syl2anc 696 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)
2819, 27jca 555 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∩ cin 3714  ∅c0 4058  𝒫 cpw 4302  {csn 4321  ⟨cop 4327  ∪ cuni 4588   class class class wbr 4804   × cxp 5264  ran crn 5267  ‘cfv 6049  1st c1st 7332  2nd c2nd 7333   ≈ cen 8120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1st 7334  df-2nd 7335  df-en 8124 This theorem is referenced by:  disjenex  8285  domss2  8286
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