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Theorem cat1 17988
Description: The definition of category df-cat 17553 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 17985 and setc2ohom 17986 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17917 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)
Assertion
Ref Expression
cat1 βˆƒπ‘ ∈ Cat [(Baseβ€˜π‘) / 𝑏][(Hom β€˜π‘) / β„Ž] Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))
Distinct variable group:   𝑏,𝑐,β„Ž,𝑀,π‘₯,𝑦,𝑧

Proof of Theorem cat1
StepHypRef Expression
1 2on 8427 . . 3 2o ∈ On
2 eqid 2733 . . . 4 (SetCatβ€˜2o) = (SetCatβ€˜2o)
32setccat 17976 . . 3 (2o ∈ On β†’ (SetCatβ€˜2o) ∈ Cat)
41, 3ax-mp 5 . 2 (SetCatβ€˜2o) ∈ Cat
51a1i 11 . . . 4 (⊀ β†’ 2o ∈ On)
6 eqid 2733 . . . 4 (Baseβ€˜(SetCatβ€˜2o)) = (Baseβ€˜(SetCatβ€˜2o))
7 eqid 2733 . . . 4 (Hom β€˜(SetCatβ€˜2o)) = (Hom β€˜(SetCatβ€˜2o))
8 0ex 5265 . . . . . . 7 βˆ… ∈ V
98prid1 4724 . . . . . 6 βˆ… ∈ {βˆ…, {βˆ…}}
10 df2o2 8422 . . . . . 6 2o = {βˆ…, {βˆ…}}
119, 10eleqtrri 2833 . . . . 5 βˆ… ∈ 2o
1211a1i 11 . . . 4 (⊀ β†’ βˆ… ∈ 2o)
13 p0ex 5340 . . . . . . 7 {βˆ…} ∈ V
1413prid2 4725 . . . . . 6 {βˆ…} ∈ {βˆ…, {βˆ…}}
1514, 10eleqtrri 2833 . . . . 5 {βˆ…} ∈ 2o
1615a1i 11 . . . 4 (⊀ β†’ {βˆ…} ∈ 2o)
17 0nep0 5314 . . . . 5 βˆ… β‰  {βˆ…}
1817a1i 11 . . . 4 (⊀ β†’ βˆ… β‰  {βˆ…})
192, 5, 6, 7, 12, 16, 18cat1lem 17987 . . 3 (⊀ β†’ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
2019mptru 1549 . 2 βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))
21 fvexd 6858 . . . 4 (𝑐 = (SetCatβ€˜2o) β†’ (Baseβ€˜π‘) ∈ V)
22 fveq2 6843 . . . 4 (𝑐 = (SetCatβ€˜2o) β†’ (Baseβ€˜π‘) = (Baseβ€˜(SetCatβ€˜2o)))
23 fvexd 6858 . . . . 5 ((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) β†’ (Hom β€˜π‘) ∈ V)
24 fveq2 6843 . . . . . 6 (𝑐 = (SetCatβ€˜2o) β†’ (Hom β€˜π‘) = (Hom β€˜(SetCatβ€˜2o)))
2524adantr 482 . . . . 5 ((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) β†’ (Hom β€˜π‘) = (Hom β€˜(SetCatβ€˜2o)))
26 oveq 7364 . . . . . . . . . . . 12 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (π‘₯β„Žπ‘¦) = (π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦))
27 oveq 7364 . . . . . . . . . . . 12 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (π‘§β„Žπ‘€) = (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀))
2826, 27ineq12d 4174 . . . . . . . . . . 11 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ ((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) = ((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)))
2928neeq1d 3000 . . . . . . . . . 10 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ↔ ((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ…))
3029anbi1d 631 . . . . . . . . 9 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ ((((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
31302rexbidv 3210 . . . . . . . 8 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
32312rexbidv 3210 . . . . . . 7 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
3332adantl 483 . . . . . 6 (((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) ∧ β„Ž = (Hom β€˜(SetCatβ€˜2o))) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
34 pm4.61 406 . . . . . . . . . . 11 (Β¬ (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
35342rexbii 3125 . . . . . . . . . 10 (βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 Β¬ (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
36 rexnal2 3129 . . . . . . . . . 10 (βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 Β¬ (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ Β¬ βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
3735, 36bitr3i 277 . . . . . . . . 9 (βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ Β¬ βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
38372rexbii 3125 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 Β¬ βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
39 rexnal2 3129 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 Β¬ βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
4038, 39bitri 275 . . . . . . 7 (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
4140a1i 11 . . . . . 6 (((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) ∧ β„Ž = (Hom β€˜(SetCatβ€˜2o))) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
42 rexeq 3309 . . . . . . . . . 10 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
43422rexbidv 3210 . . . . . . . . 9 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
4443rexbidv 3172 . . . . . . . 8 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
45 rexeq 3309 . . . . . . . . 9 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
46452rexbidv 3210 . . . . . . . 8 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
47 rexeq 3309 . . . . . . . . 9 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
4847rexeqbi1dv 3307 . . . . . . . 8 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
4944, 46, 483bitrd 305 . . . . . . 7 (𝑏 = (Baseβ€˜(SetCatβ€˜2o)) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
5049ad2antlr 726 . . . . . 6 (((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) ∧ β„Ž = (Hom β€˜(SetCatβ€˜2o))) β†’ (βˆƒπ‘₯ ∈ 𝑏 βˆƒπ‘¦ ∈ 𝑏 βˆƒπ‘§ ∈ 𝑏 βˆƒπ‘€ ∈ 𝑏 (((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
5133, 41, 503bitr3d 309 . . . . 5 (((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) ∧ β„Ž = (Hom β€˜(SetCatβ€˜2o))) β†’ (Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
5223, 25, 51sbcied2 3787 . . . 4 ((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) β†’ ([(Hom β€˜π‘) / β„Ž] Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
5321, 22, 52sbcied2 3787 . . 3 (𝑐 = (SetCatβ€˜2o) β†’ ([(Baseβ€˜π‘) / 𝑏][(Hom β€˜π‘) / β„Ž] Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)) ↔ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))))
5453rspcev 3580 . 2 (((SetCatβ€˜2o) ∈ Cat ∧ βˆƒπ‘₯ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘¦ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘§ ∈ (Baseβ€˜(SetCatβ€˜2o))βˆƒπ‘€ ∈ (Baseβ€˜(SetCatβ€˜2o))(((π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦) ∩ (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀)) β‰  βˆ… ∧ Β¬ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))) β†’ βˆƒπ‘ ∈ Cat [(Baseβ€˜π‘) / 𝑏][(Hom β€˜π‘) / β„Ž] Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀)))
554, 20, 54mp2an 691 1 βˆƒπ‘ ∈ Cat [(Baseβ€˜π‘) / 𝑏][(Hom β€˜π‘) / β„Ž] Β¬ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 βˆ€π‘§ ∈ 𝑏 βˆ€π‘€ ∈ 𝑏 (((π‘₯β„Žπ‘¦) ∩ (π‘§β„Žπ‘€)) β‰  βˆ… β†’ (π‘₯ = 𝑧 ∧ 𝑦 = 𝑀))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βŠ€wtru 1543   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444  [wsbc 3740   ∩ cin 3910  βˆ…c0 4283  {csn 4587  {cpr 4589  Oncon0 6318  β€˜cfv 6497  (class class class)co 7358  2oc2o 8407  Basecbs 17088  Hom chom 17149  Catccat 17549  SetCatcsetc 17966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-setc 17967
This theorem is referenced by: (None)
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