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Theorem brprcneu 5198
Description: If 𝐴 is a proper class, then there is no unique binary relationship with 𝐴 as the first element. (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 dtruex 4310 . . . . . . . . 9 𝑦 ¬ 𝑦 = 𝑥
2 equcom 1609 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
32notbii 604 . . . . . . . . . 10 𝑥 = 𝑦 ↔ ¬ 𝑦 = 𝑥)
43exbii 1512 . . . . . . . . 9 (∃𝑦 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ¬ 𝑦 = 𝑥)
51, 4mpbir 138 . . . . . . . 8 𝑦 ¬ 𝑥 = 𝑦
65jctr 302 . . . . . . 7 (∅ ∈ 𝐹 → (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
7 19.42v 1802 . . . . . . 7 (∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ∃𝑦 ¬ 𝑥 = 𝑦))
86, 7sylibr 141 . . . . . 6 (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))
9 opprc1 3598 . . . . . . . 8 𝐴 ∈ V → ⟨𝐴, 𝑥⟩ = ∅)
109eleq1d 2122 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
11 opprc1 3598 . . . . . . . . . . . 12 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
1211eleq1d 2122 . . . . . . . . . . 11 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝐹 ↔ ∅ ∈ 𝐹))
1310, 12anbi12d 450 . . . . . . . . . 10 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ (∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹)))
14 anidm 382 . . . . . . . . . 10 ((∅ ∈ 𝐹 ∧ ∅ ∈ 𝐹) ↔ ∅ ∈ 𝐹)
1513, 14syl6bb 189 . . . . . . . . 9 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ↔ ∅ ∈ 𝐹))
1615anbi1d 446 . . . . . . . 8 𝐴 ∈ V → (((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ (∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
1716exbidv 1722 . . . . . . 7 𝐴 ∈ V → (∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦)))
1810, 17imbi12d 227 . . . . . 6 𝐴 ∈ V → ((⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)) ↔ (∅ ∈ 𝐹 → ∃𝑦(∅ ∈ 𝐹 ∧ ¬ 𝑥 = 𝑦))))
198, 18mpbiri 161 . . . . 5 𝐴 ∈ V → (⟨𝐴, 𝑥⟩ ∈ 𝐹 → ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦)))
20 df-br 3792 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
21 df-br 3792 . . . . . . . 8 (𝐴𝐹𝑦 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹)
2220, 21anbi12i 441 . . . . . . 7 ((𝐴𝐹𝑥𝐴𝐹𝑦) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
2322anbi1i 439 . . . . . 6 (((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
2423exbii 1512 . . . . 5 (∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) ↔ ∃𝑦((⟨𝐴, 𝑥⟩ ∈ 𝐹 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝐹) ∧ ¬ 𝑥 = 𝑦))
2519, 20, 243imtr4g 198 . . . 4 𝐴 ∈ V → (𝐴𝐹𝑥 → ∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
2625eximdv 1776 . . 3 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦)))
27 exanaliim 1554 . . . . . 6 (∃𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) → ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
2827eximi 1507 . . . . 5 (∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) → ∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
29 exnalim 1553 . . . . 5 (∃𝑥 ¬ ∀𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦) → ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3028, 29syl 14 . . . 4 (∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) → ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
31 breq2 3795 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝐹𝑥𝐴𝐹𝑦))
3231mo4 1977 . . . . 5 (∃*𝑥 𝐴𝐹𝑥 ↔ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3332notbii 604 . . . 4 (¬ ∃*𝑥 𝐴𝐹𝑥 ↔ ¬ ∀𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) → 𝑥 = 𝑦))
3430, 33sylibr 141 . . 3 (∃𝑥𝑦((𝐴𝐹𝑥𝐴𝐹𝑦) ∧ ¬ 𝑥 = 𝑦) → ¬ ∃*𝑥 𝐴𝐹𝑥)
3526, 34syl6 33 . 2 𝐴 ∈ V → (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
36 eu5 1963 . . . 4 (∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
3736notbii 604 . . 3 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
38 imnan 634 . . 3 ((∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥) ↔ ¬ (∃𝑥 𝐴𝐹𝑥 ∧ ∃*𝑥 𝐴𝐹𝑥))
3937, 38bitr4i 180 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (∃𝑥 𝐴𝐹𝑥 → ¬ ∃*𝑥 𝐴𝐹𝑥))
4035, 39sylibr 141 1 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wal 1257  wex 1397  wcel 1409  ∃!weu 1916  ∃*wmo 1917  Vcvv 2574  c0 3251  cop 3405   class class class wbr 3791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792
This theorem is referenced by:  fvprc  5199
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