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Theorem brprcneu 6655
Description: If 𝐴 is a proper class and 𝐹 is any class, then there is no unique set which is related to 𝐴 through the binary relation 𝐹. (Contributed by Scott Fenton, 7-Oct-2017.)
Assertion
Ref Expression
brprcneu 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem brprcneu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dtru 5262 . . . . . . . . 9 ¬ ∀𝑦 𝑦 = 𝑥
2 exnal 1821 . . . . . . . . . 10 (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑥 = 𝑦)
3 equcom 2019 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
43albii 1814 . . . . . . . . . 10 (∀𝑦 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
52, 4xchbinx 336 . . . . . . . . 9 (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑦 = 𝑥)
61, 5mpbir 233 . . . . . . . 8 𝑦 ¬ 𝑥 = 𝑦
76jctr 527 . . . . . . 7 (∅ ∈ 𝐹 → (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
8 19.42v 1948 . . . . . . 7 (∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
97, 8sylibr 236 . . . . . 6 (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))
10 opprc1 4819 . . . . . . . 8 𝐴 ∈ V → ⟨𝐴, 𝑥⟩ = ∅)
1110eleq1d 2895 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
12 opprc1 4819 . . . . . . . . . . . 12 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
1312eleq1d 2895 . . . . . . . . . . 11 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
1411, 13anbi12d 632 . . . . . . . . . 10 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹)))
15 anidm 567 . . . . . . . . . 10 ((∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹) ↔ ∅ ∈ 𝐹)
1614, 15syl6bb 289 . . . . . . . . 9 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ ∅ ∈ 𝐹))
1716anbi1d 631 . . . . . . . 8 𝐴 ∈ V → (((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
1817exbidv 1916 . . . . . . 7 𝐴 ∈ V → (∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
1911, 18imbi12d 347 . . . . . 6 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)) ↔ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))))
209, 19mpbiri 260 . . . . 5 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)))
21 df-br 5058 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
22 df-br 5058 . . . . . . . 8 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
2321, 22anbi12i 628 . . . . . . 7 ((𝐴𝐹𝑥𝐴𝐹𝑦) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2423anbi1i 625 . . . . . 6 (((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
2524exbii 1842 . . . . 5 (∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
2620, 21, 253imtr4g 298 . . . 4 𝐴 ∈ V → (𝐴𝐹𝑥 → ∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
2726eximdv 1912 . . 3 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
28 exnal 1821 . . . 4 (∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
29 exanali 1853 . . . . 5 (∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3029exbii 1842 . . . 4 (∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
31 breq2 5061 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝐹𝑥𝐴𝐹𝑦))
3231mo4 2644 . . . . 5 (∃*𝑥 𝐴𝐹𝑥 ↔ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3332notbii 322 . . . 4 (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3428, 30, 333bitr4ri 306 . . 3 (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
3527, 34syl6ibr 254 . 2 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
36 df-eu 2648 . . . 4 (∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
3736notbii 322 . . 3 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
38 imnan 402 . . 3 ((∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥) ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
3937, 38bitr4i 280 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
4035, 39sylibr 236 1 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1529  wex 1774  wcel 2108  ∃*wmo 2614  ∃!weu 2647  Vcvv 3493  c0 4289  cop 4565   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201  ax-pow 5257
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058
This theorem is referenced by:  fvprc  6656  eubrv  43261
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