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Theorem kur14 32707
 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 2839 . . . . . . . . 9 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝐴𝑥 ↔ if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥))
32anbi1d 632 . . . . . . . 8 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)))
43rabbidv 3392 . . . . . . 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 4846 . . . . . 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 2805 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
76eleq1d 2836 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin))
86fveq2d 6667 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (♯‘𝑆) = (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}))
98breq1d 5046 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((♯‘𝑆) ≤ 14 ↔ (♯‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14))
107, 9anbi12d 633 . . 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 4812 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}))
1311, 12syl5eq 2805 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = if(𝐽 ∈ Top, 𝐽, {∅}))
1413pweqd 4516 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1514pweqd 4516 . . . . . . 7 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1613sseq2d 3926 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 if(𝐽 ∈ Top, 𝐽, {∅})))
17 sn0top 21713 . . . . . . . . . . . . . 14 {∅} ∈ Top
1817elimel 4492 . . . . . . . . . . . . 13 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19 uniexg 7470 . . . . . . . . . . . . 13 (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
2018, 19ax-mp 5 . . . . . . . . . . . 12 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
2120elpw2 5219 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 if(𝐽 ∈ Top, 𝐽, {∅}))
2216, 21bitr4di 292 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})))
2322ifbid 4446 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
2423eleq1d 2836 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
2513difeq1d 4029 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋𝑦) = ( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26 kur14.k . . . . . . . . . . . . 13 𝐾 = (cls‘𝐽)
27 fveq2 6663 . . . . . . . . . . . . 13 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2826, 27syl5eq 2805 . . . . . . . . . . . 12 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2928fveq1d 6665 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
3025, 29preq12d 4637 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋𝑦), (𝐾𝑦)} = {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
3130sseq1d 3925 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
3231ralbidv 3126 . . . . . . . 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 633 . . . . . . 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 3398 . . . . . 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 4846 . . . . 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 2836 . . . 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 6667 . . . . 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 5046 . . . 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 633 . . 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 2758 . . . 4 if(𝐽 ∈ Top, 𝐽, {∅}) = if(𝐽 ∈ Top, 𝐽, {∅})
41 eqid 2758 . . . 4 (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42 eqid 2758 . . . 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 5228 . . . . . 6 ∅ ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
4443elimel 4492 . . . . 5 if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
45 elpwi 4506 . . . . 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 32706 . . 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 4482 . 2 ((𝐴𝑋𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
4948ancoms 462 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3857   ⊆ wss 3860  ∅c0 4227  ifcif 4423  𝒫 cpw 4497  {csn 4525  {cpr 4527  ∪ cuni 4801  ∩ cint 4841   class class class wbr 5036  ‘cfv 6340  Fincfn 8540  1c1 10589   ≤ cle 10727  4c4 11744  ;cdc 12150  ♯chash 13753  Topctop 21607  clsccl 21732 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-oadd 8122  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-dju 9376  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-xnn0 12020  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-hash 13754  df-top 21608  df-topon 21625  df-cld 21733  df-ntr 21734  df-cls 21735 This theorem is referenced by: (None)
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