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Theorem brprcneuALT 6833
Description: Alternate proof of brprcneu 6832 using ax-pow 5320 instead of ax-pr 5384. (Contributed by Scott Fenton, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
brprcneuALT 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem brprcneuALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dtruALT2 5325 . . . . . . . . 9 ¬ ∀𝑦 𝑦 = 𝑥
2 exnal 1829 . . . . . . . . . 10 (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑥 = 𝑦)
3 equcom 2021 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
43albii 1821 . . . . . . . . . 10 (∀𝑦 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
52, 4xchbinx 333 . . . . . . . . 9 (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑦 = 𝑥)
61, 5mpbir 230 . . . . . . . 8 𝑦 ¬ 𝑥 = 𝑦
76jctr 525 . . . . . . 7 (∅ ∈ 𝐹 → (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
8 19.42v 1957 . . . . . . 7 (∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
97, 8sylibr 233 . . . . . 6 (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))
10 opprc1 4854 . . . . . . . 8 𝐴 ∈ V → ⟨𝐴, 𝑥⟩ = ∅)
1110eleq1d 2822 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
12 opprc1 4854 . . . . . . . . . . . 12 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
1312eleq1d 2822 . . . . . . . . . . 11 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
1411, 13anbi12d 631 . . . . . . . . . 10 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹)))
15 anidm 565 . . . . . . . . . 10 ((∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹) ↔ ∅ ∈ 𝐹)
1614, 15bitrdi 286 . . . . . . . . 9 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ ∅ ∈ 𝐹))
1716anbi1d 630 . . . . . . . 8 𝐴 ∈ V → (((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
1817exbidv 1924 . . . . . . 7 𝐴 ∈ V → (∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
1911, 18imbi12d 344 . . . . . 6 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)) ↔ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))))
209, 19mpbiri 257 . . . . 5 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)))
21 df-br 5106 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
22 df-br 5106 . . . . . . . 8 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
2321, 22anbi12i 627 . . . . . . 7 ((𝐴𝐹𝑥𝐴𝐹𝑦) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2423anbi1i 624 . . . . . 6 (((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
2524exbii 1850 . . . . 5 (∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
2620, 21, 253imtr4g 295 . . . 4 𝐴 ∈ V → (𝐴𝐹𝑥 → ∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
2726eximdv 1920 . . 3 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
28 exnal 1829 . . . 4 (∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
29 exanali 1862 . . . . 5 (∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3029exbii 1850 . . . 4 (∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
31 breq2 5109 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝐹𝑥𝐴𝐹𝑦))
3231mo4 2564 . . . . 5 (∃*𝑥 𝐴𝐹𝑥 ↔ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3332notbii 319 . . . 4 (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3428, 30, 333bitr4ri 303 . . 3 (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
3527, 34syl6ibr 251 . 2 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
36 df-eu 2567 . . . 4 (∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
3736notbii 319 . . 3 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
38 imnan 400 . . 3 ((∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥) ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
3937, 38bitr4i 277 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
4035, 39sylibr 233 1 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1539  wex 1781  wcel 2106  ∃*wmo 2536  ∃!weu 2566  Vcvv 3445  c0 4282  cop 4592   class class class wbr 5105
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263  ax-pow 5320
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106
This theorem is referenced by:  fvprcALT  6835
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