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Theorem cat1 18079
Description: The definition of category df-cat 17641 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18076 and setc2ohom 18077 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18008 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 8494 . . 3 2o ∈ On
2 eqid 2728 . . . 4 (SetCatβ€˜2o) = (SetCatβ€˜2o)
32setccat 18067 . . 3 (2o ∈ On β†’ (SetCatβ€˜2o) ∈ Cat)
41, 3ax-mp 5 . 2 (SetCatβ€˜2o) ∈ Cat
51a1i 11 . . . 4 (⊀ β†’ 2o ∈ On)
6 eqid 2728 . . . 4 (Baseβ€˜(SetCatβ€˜2o)) = (Baseβ€˜(SetCatβ€˜2o))
7 eqid 2728 . . . 4 (Hom β€˜(SetCatβ€˜2o)) = (Hom β€˜(SetCatβ€˜2o))
8 0ex 5301 . . . . . . 7 βˆ… ∈ V
98prid1 4762 . . . . . 6 βˆ… ∈ {βˆ…, {βˆ…}}
10 df2o2 8489 . . . . . 6 2o = {βˆ…, {βˆ…}}
119, 10eleqtrri 2828 . . . . 5 βˆ… ∈ 2o
1211a1i 11 . . . 4 (⊀ β†’ βˆ… ∈ 2o)
13 p0ex 5378 . . . . . . 7 {βˆ…} ∈ V
1413prid2 4763 . . . . . 6 {βˆ…} ∈ {βˆ…, {βˆ…}}
1514, 10eleqtrri 2828 . . . . 5 {βˆ…} ∈ 2o
1615a1i 11 . . . 4 (⊀ β†’ {βˆ…} ∈ 2o)
17 0nep0 5352 . . . . 5 βˆ… β‰  {βˆ…}
1817a1i 11 . . . 4 (⊀ β†’ βˆ… β‰  {βˆ…})
192, 5, 6, 7, 12, 16, 18cat1lem 18078 . . 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 1541 . 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 6906 . . . 4 (𝑐 = (SetCatβ€˜2o) β†’ (Baseβ€˜π‘) ∈ V)
22 fveq2 6891 . . . 4 (𝑐 = (SetCatβ€˜2o) β†’ (Baseβ€˜π‘) = (Baseβ€˜(SetCatβ€˜2o)))
23 fvexd 6906 . . . . 5 ((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) β†’ (Hom β€˜π‘) ∈ V)
24 fveq2 6891 . . . . . 6 (𝑐 = (SetCatβ€˜2o) β†’ (Hom β€˜π‘) = (Hom β€˜(SetCatβ€˜2o)))
2524adantr 480 . . . . 5 ((𝑐 = (SetCatβ€˜2o) ∧ 𝑏 = (Baseβ€˜(SetCatβ€˜2o))) β†’ (Hom β€˜π‘) = (Hom β€˜(SetCatβ€˜2o)))
26 oveq 7420 . . . . . . . . . . . 12 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (π‘₯β„Žπ‘¦) = (π‘₯(Hom β€˜(SetCatβ€˜2o))𝑦))
27 oveq 7420 . . . . . . . . . . . 12 (β„Ž = (Hom β€˜(SetCatβ€˜2o)) β†’ (π‘§β„Žπ‘€) = (𝑧(Hom β€˜(SetCatβ€˜2o))𝑀))
2826, 27ineq12d 4209 . . . . . . . . . . 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 2996 . . . . . . . . . 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 630 . . . . . . . . 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 3215 . . . . . . . 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 3215 . . . . . . 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 481 . . . . . 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 pm4.61 404 . . . . . . . . . . 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 3131 . . . . . . . . . 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 3131 . . . . . . . 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 3317 . . . . . . . . . 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 3215 . . . . . . . . 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 3174 . . . . . . . 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 3317 . . . . . . . . 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 3215 . . . . . . . 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 3317 . . . . . . . . 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 3330 . . . . . . . 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 3822 . . . 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 3822 . . 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 3608 . 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 395   = wceq 1534  βŠ€wtru 1535   ∈ wcel 2099   β‰  wne 2936  βˆ€wral 3057  βˆƒwrex 3066  Vcvv 3470  [wsbc 3775   ∩ cin 3944  βˆ…c0 4318  {csn 4624  {cpr 4626  Oncon0 6363  β€˜cfv 6542  (class class class)co 7414  2oc2o 8474  Basecbs 17173  Hom chom 17237  Catccat 17637  SetCatcsetc 18057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12497  df-z 12583  df-dec 12702  df-uz 12847  df-fz 13511  df-struct 17109  df-slot 17144  df-ndx 17156  df-base 17174  df-hom 17250  df-cco 17251  df-cat 17641  df-cid 17642  df-setc 18058
This theorem is referenced by: (None)
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