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Theorem kur14 31667
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 2832 . . . . . . . . 9 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝐴𝑥 ↔ if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥))
32anbi1d 623 . . . . . . . 8 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)))
43rabbidv 3338 . . . . . . 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 4640 . . . . . 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 2811 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
76eleq1d 2829 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin))
86fveq2d 6383 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (♯‘𝑆) = (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}))
98breq1d 4821 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((♯‘𝑆) ≤ 14 ↔ (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14))
107, 9anbi12d 624 . . 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 4604 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}))
1311, 12syl5eq 2811 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = if(𝐽 ∈ Top, 𝐽, {∅}))
1413pweqd 4322 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1514pweqd 4322 . . . . . . 7 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1613sseq2d 3795 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 if(𝐽 ∈ Top, 𝐽, {∅})))
17 sn0top 21097 . . . . . . . . . . . . . 14 {∅} ∈ Top
1817elimel 4312 . . . . . . . . . . . . 13 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19 uniexg 7157 . . . . . . . . . . . . 13 (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
2018, 19ax-mp 5 . . . . . . . . . . . 12 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
2120elpw2 4988 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 if(𝐽 ∈ Top, 𝐽, {∅}))
2216, 21syl6bbr 280 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})))
2322ifbid 4267 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
2423eleq1d 2829 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
2513difeq1d 3891 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋𝑦) = ( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26 kur14.k . . . . . . . . . . . . 13 𝐾 = (cls‘𝐽)
27 fveq2 6379 . . . . . . . . . . . . 13 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2826, 27syl5eq 2811 . . . . . . . . . . . 12 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2928fveq1d 6381 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
3025, 29preq12d 4433 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋𝑦), (𝐾𝑦)} = {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
3130sseq1d 3794 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
3231ralbidv 3133 . . . . . . . 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 624 . . . . . . 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 3344 . . . . . 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 4640 . . . . 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 2829 . . . 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 6383 . . . . 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 4821 . . . 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 624 . . 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 2765 . . . 4 if(𝐽 ∈ Top, 𝐽, {∅}) = if(𝐽 ∈ Top, 𝐽, {∅})
41 eqid 2765 . . . 4 (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42 eqid 2765 . . . 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 4994 . . . . . 6 ∅ ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
4443elimel 4312 . . . . 5 if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
45 elpwi 4327 . . . . 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 31666 . . 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 4302 . 2 ((𝐴𝑋𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
4948ancoms 450 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  c0 4081  ifcif 4245  𝒫 cpw 4317  {csn 4336  {cpr 4338   cuni 4596   cint 4635   class class class wbr 4811  cfv 6070  Fincfn 8164  1c1 10194  cle 10333  4c4 11333  cdc 11745  chash 13326  Topctop 20991  clsccl 21116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-3 11340  df-4 11341  df-5 11342  df-6 11343  df-7 11344  df-8 11345  df-9 11346  df-n0 11543  df-xnn0 11615  df-z 11629  df-dec 11746  df-uz 11892  df-fz 12539  df-hash 13327  df-top 20992  df-topon 21009  df-cld 21117  df-ntr 21118  df-cls 21119
This theorem is referenced by: (None)
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