Theorem brprcneu 6897

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 6898 for a proof that uses ax-pow 5371 instead of ax-pr 5438. (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 5447	. . . . . . . . 9 ⊢ ¬ ∀𝑦 𝑦 = 𝑥
2		exnal 1824	. . . . . . . . . 10 ⊢ (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑥 = 𝑦)
3		equcom 2015	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
4	3	albii 1816	. . . . . . . . . 10 ⊢ (∀𝑦 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
5	2, 4	xchbinx 334	. . . . . . . . 9 ⊢ (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 𝑦 = 𝑥)
6	1, 5	mpbir 231	. . . . . . . 8 ⊢ ∃𝑦 ¬ 𝑥 = 𝑦
7	6	jctr 524	. . . . . . 7 ⊢ (∅ ∈ 𝐹 → (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
8		19.42v 1951	. . . . . . 7 ⊢ (∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
9	7, 8	sylibr 234	. . . . . 6 ⊢ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))
10		opprc1 4902	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑥⟩ = ∅)
11	10	eleq1d 2824	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
12		opprc1 4902	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
13	12	eleq1d 2824	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
14	11, 13	anbi12d 632	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹)))
15		anidm 564	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹) ↔ ∅ ∈ 𝐹)
16	14, 15	bitrdi 287	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ ∅ ∈ 𝐹))
17	16	anbi1d 631	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
18	17	exbidv 1919	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
19	11, 18	imbi12d 344	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)) ↔ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))))
20	9, 19	mpbiri 258	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)))
21		df-br 5149	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
22		df-br 5149	. . . . . . . 8 ⊢ (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
23	21, 22	anbi12i 628	. . . . . . 7 ⊢ ((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
24	23	anbi1i 624	. . . . . 6 ⊢ (((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
25	24	exbii 1845	. . . . 5 ⊢ (∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
26	20, 21, 25	3imtr4g 296	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝑥 → ∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
27	26	eximdv 1915	. . 3 ⊢ (¬ 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
28		exnal 1824	. . . 4 ⊢ (∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
29		exanali 1857	. . . . 5 ⊢ (∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
30	29	exbii 1845	. . . 4 ⊢ (∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
31		breq2 5152	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦))
32	31	mo4 2564	. . . . 5 ⊢ (∃*𝑥 𝐴𝐹𝑥 ↔ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
33	32	notbii 320	. . . 4 ⊢ (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ¬ ∀𝑥∀𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) → 𝑥 = 𝑦))
34	28, 30, 33	3bitr4ri 304	. . 3 ⊢ (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ∃𝑥∃𝑦((𝐴𝐹𝑥 ∧ 𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦))
35	27, 34	imbitrrdi 252	. 2 ⊢ (¬ 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
36		df-eu 2567	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
37	36	notbii 320	. . 3 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
38		imnan 399	. . 3 ⊢ ((∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥) ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
39	37, 38	bitr4i 278	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
40	35, 39	sylibr 234	1 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1776   ∈ wcel 2106  ∃*wmo 2536  ∃!weu 2566  Vcvv 3478  ∅c0 4339  ⟨cop 4637   class class class wbr 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149

This theorem is referenced by:  fvprc  6899  eubrv  46985