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Theorem kur14 32350
Description: Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14.x 𝑋 = 𝐽
kur14.k 𝐾 = (cls‘𝐽)
kur14.s 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}
Assertion
Ref Expression
kur14 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem kur14
StepHypRef Expression
1 kur14.s . . . . . 6 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}
2 eleq1 2905 . . . . . . . . 9 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝐴𝑥 ↔ if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥))
32anbi1d 629 . . . . . . . 8 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)))
43rabbidv 3486 . . . . . . 7 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
54inteqd 4879 . . . . . 6 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
61, 5syl5eq 2873 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
76eleq1d 2902 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin))
86fveq2d 6671 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (♯‘𝑆) = (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}))
98breq1d 5073 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((♯‘𝑆) ≤ 14 ↔ (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14))
107, 9anbi12d 630 . . 3 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14) ↔ ( {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14)))
11 kur14.x . . . . . . . . . 10 𝑋 = 𝐽
12 unieq 4845 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}))
1311, 12syl5eq 2873 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = if(𝐽 ∈ Top, 𝐽, {∅}))
1413pweqd 4547 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1514pweqd 4547 . . . . . . 7 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1613sseq2d 4003 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 if(𝐽 ∈ Top, 𝐽, {∅})))
17 sn0top 21526 . . . . . . . . . . . . . 14 {∅} ∈ Top
1817elimel 4537 . . . . . . . . . . . . 13 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19 uniexg 7457 . . . . . . . . . . . . 13 (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
2018, 19ax-mp 5 . . . . . . . . . . . 12 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
2120elpw2 5245 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 if(𝐽 ∈ Top, 𝐽, {∅}))
2216, 21syl6bbr 290 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})))
2322ifbid 4492 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
2423eleq1d 2902 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
2513difeq1d 4102 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋𝑦) = ( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26 kur14.k . . . . . . . . . . . . 13 𝐾 = (cls‘𝐽)
27 fveq2 6667 . . . . . . . . . . . . 13 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2826, 27syl5eq 2873 . . . . . . . . . . . 12 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2928fveq1d 6669 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
3025, 29preq12d 4676 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋𝑦), (𝐾𝑦)} = {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
3130sseq1d 4002 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
3231ralbidv 3202 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
3324, 32anbi12d 630 . . . . . . 7 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)))
3415, 33rabeqbidv 3491 . . . . . 6 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)})
3534inteqd 4879 . . . . 5 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)})
3635eleq1d 2902 . . . 4 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ( {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin ↔ {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin))
3735fveq2d 6671 . . . . 5 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) = (♯‘ {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}))
3837breq1d 5073 . . . 4 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14 ↔ (♯‘ {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ 14))
3936, 38anbi12d 630 . . 3 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (( {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14) ↔ ( {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘ {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ 14)))
40 eqid 2826 . . . 4 if(𝐽 ∈ Top, 𝐽, {∅}) = if(𝐽 ∈ Top, 𝐽, {∅})
41 eqid 2826 . . . 4 (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42 eqid 2826 . . . 4 {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}
43 0elpw 5253 . . . . . 6 ∅ ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
4443elimel 4537 . . . . 5 if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
45 elpwi 4554 . . . . 5 (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ⊆ if(𝐽 ∈ Top, 𝐽, {∅}))
4644, 45ax-mp 5 . . . 4 if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ⊆ if(𝐽 ∈ Top, 𝐽, {∅})
4718, 40, 41, 42, 46kur14lem10 32349 . . 3 ( {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘ {𝑥 ∈ 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ 14)
4810, 39, 47dedth2h 4527 . 2 ((𝐴𝑋𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
4948ancoms 459 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cdif 3937  wss 3940  c0 4295  ifcif 4470  𝒫 cpw 4542  {csn 4564  {cpr 4566   cuni 4837   cint 4874   class class class wbr 5063  cfv 6352  Fincfn 8498  1c1 10527  cle 10665  4c4 11683  cdc 12087  chash 13680  Topctop 21420  clsccl 21545
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-xnn0 11957  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-hash 13681  df-top 21421  df-topon 21438  df-cld 21546  df-ntr 21547  df-cls 21548
This theorem is referenced by: (None)
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