Theorem brprcneu 6886

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. See brprcneuALT 6887 for a proof that uses ax-pow 5365 instead of ax-pr 5429. (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.

Step	Hyp	Ref	Expression
1		dtru 5438	. . . . . . . . 9 ⊢ ¬ ∀𝑦 𝑦 = 𝑥
2		exnal 1821	. . . . . . . . . 10 ⊢ (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑥 = 𝑦)
3		equcom 2013	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
4	3	albii 1813	. . . . . . . . . 10 ⊢ (∀𝑦 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
5	2, 4	xchbinx 333	. . . . . . . . 9 ⊢ (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑦 = 𝑥)
6	1, 5	mpbir 230	. . . . . . . 8 ⊢ ∃𝑦 ¬ 𝑥 = 𝑦
7	6	jctr 523	. . . . . . 7 ⊢ (∅ ∈ 𝐹 → (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
8		19.42v 1949	. . . . . . 7 ⊢ (∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
9	7, 8	sylibr 233	. . . . . 6 ⊢ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))
10		opprc1 4899	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑥⟩ = ∅)
11	10	eleq1d 2810	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
12		opprc1 4899	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
13	12	eleq1d 2810	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
14	11, 13	anbi12d 630	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹)))
15		anidm 563	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹) ↔ ∅ ∈ 𝐹)
16	14, 15	bitrdi 286	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ ∅ ∈ 𝐹))
17	16	anbi1d 629	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
18	17	exbidv 1916	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
19	11, 18	imbi12d 343	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)) ↔ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))))
20	9, 19	mpbiri 257	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)))
21		df-br 5150	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
22		df-br 5150	. . . . . . . 8 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
23	21, 22	anbi12i 626	. . . . . . 7 ⊢ ((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
24	23	anbi1i 622	. . . . . 6 ⊢ (((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
25	24	exbii 1842	. . . . 5 ⊢ (∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
26	20, 21, 25	3imtr4g 295	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝑥 → ∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
27	26	eximdv 1912	. . 3 ⊢ (¬ 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
28		exnal 1821	. . . 4 ⊢ (∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
29		exanali 1854	. . . . 5 ⊢ (∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
30	29	exbii 1842	. . . 4 ⊢ (∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
31		breq2 5153	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦))
32	31	mo4 2554	. . . . 5 ⊢ (∃*𝑥 𝐴𝐹𝑥 ↔ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
33	32	notbii 319	. . . 4 ⊢ (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ¬ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
34	28, 30, 33	3bitr4ri 303	. . 3 ⊢ (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
35	27, 34	imbitrrdi 251	. 2 ⊢ (¬ 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
36		df-eu 2557	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
37	36	notbii 319	. . 3 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
38		imnan 398	. . 3 ⊢ ((∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥) ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
39	37, 38	bitr4i 277	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
40	35, 39	sylibr 233	1 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394  ∀wal 1531  ∃wex 1773   ∈ wcel 2098  ∃*wmo 2526  ∃!weu 2556  Vcvv 3461  ∅c0 4322  ⟨cop 4636   class class class wbr 5149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150

This theorem is referenced by:  fvprc  6888  eubrv  46552