Theorem brprcneuALT 6872

Description: Alternate proof of brprcneu 6871 using ax-pow 5336 instead of ax-pr 5404. (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.

Step	Hyp	Ref	Expression
1		dtruALT2 5341	. . . . . . . . 9 ⊢ ¬ ∀𝑦 𝑦 = 𝑥
2		exnal 1857	. . . . . . . . . 10 ⊢ (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑥 = 𝑦)
3		equcom 2048	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
4	3	albii 1849	. . . . . . . . . 10 ⊢ (∀𝑦 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
5	2, 4	xchbinx 337	. . . . . . . . 9 ⊢ (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑦 = 𝑥)
6	1, 5	mpbir 234	. . . . . . . 8 ⊢ ∃𝑦 ¬ 𝑥 = 𝑦
7	6	jctr 533	. . . . . . 7 ⊢ (∅ ∈ 𝐹 → (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
8		19.42v 1983	. . . . . . 7 ⊢ (∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
9	7, 8	sylibr 237	. . . . . 6 ⊢ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))
10		opprc1 4862	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑥⟩ = ∅)
11	10	eleq1d 2848	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
12		opprc1 4862	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
13	12	eleq1d 2848	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
14	11, 13	anbi12d 643	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹)))
15		anidm 574	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹) ↔ ∅ ∈ 𝐹)
16	14, 15	bitrdi 290	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ ∅ ∈ 𝐹))
17	16	anbi1d 642	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
18	17	exbidv 1951	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
19	11, 18	imbi12d 347	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)) ↔ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))))
20	9, 19	mpbiri 261	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)))
21		df-br 5110	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
22		df-br 5110	. . . . . . . 8 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
23	21, 22	anbi12i 639	. . . . . . 7 ⊢ ((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
24	23	anbi1i 635	. . . . . 6 ⊢ (((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
25	24	exbii 1878	. . . . 5 ⊢ (∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
26	20, 21, 25	3imtr4g 299	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝑥 → ∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
27	26	eximdv 1947	. . 3 ⊢ (¬ 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
28		exnal 1857	. . . 4 ⊢ (∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
29		exanali 1889	. . . . 5 ⊢ (∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
30	29	exbii 1878	. . . 4 ⊢ (∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
31		breq2 5113	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦))
32	31	mo4 2594	. . . . 5 ⊢ (∃*𝑥 𝐴𝐹𝑥 ↔ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
33	32	notbii 323	. . . 4 ⊢ (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ¬ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
34	28, 30, 33	3bitr4ri 307	. . 3 ⊢ (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
35	27, 34	imbitrrdi 255	. 2 ⊢ (¬ 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
36		df-eu 2597	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
37	36	notbii 323	. . 3 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
38		imnan 404	. . 3 ⊢ ((∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥) ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
39	37, 38	bitr4i 281	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
40	35, 39	sylibr 237	1 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565  ∃!weu 2596  Vcvv 3455  ∅c0 4286  ⟨cop 4595   class class class wbr 5109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110

This theorem is referenced by:  fvprcALT  6874