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Theorem kur14 30452
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 2676 . . . . . . . . 9 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝐴𝑥 ↔ if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥))
32anbi1d 737 . . . . . . . 8 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)))
43rabbidv 3164 . . . . . . 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 4415 . . . . . 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 2656 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
76eleq1d 2672 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin))
86fveq2d 6107 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (#‘𝑆) = (#‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}))
98breq1d 4593 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((#‘𝑆) ≤ 14 ↔ (#‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14))
107, 9anbi12d 743 . . 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 4380 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}))
1311, 12syl5eq 2656 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = if(𝐽 ∈ Top, 𝐽, {∅}))
1413pweqd 4113 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1514pweqd 4113 . . . . . . 7 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1613sseq2d 3596 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 if(𝐽 ∈ Top, 𝐽, {∅})))
17 sn0top 20613 . . . . . . . . . . . . . 14 {∅} ∈ Top
1817elimel 4100 . . . . . . . . . . . . 13 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19 uniexg 6853 . . . . . . . . . . . . 13 (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
2018, 19ax-mp 5 . . . . . . . . . . . 12 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
2120elpw2 4755 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 if(𝐽 ∈ Top, 𝐽, {∅}))
2216, 21syl6bbr 277 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})))
2322ifbid 4058 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
2423eleq1d 2672 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
2513difeq1d 3689 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋𝑦) = ( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26 kur14.k . . . . . . . . . . . . 13 𝐾 = (cls‘𝐽)
27 fveq2 6103 . . . . . . . . . . . . 13 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2826, 27syl5eq 2656 . . . . . . . . . . . 12 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2928fveq1d 6105 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
3025, 29preq12d 4220 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋𝑦), (𝐾𝑦)} = {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
3130sseq1d 3595 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
3231ralbidv 2969 . . . . . . . 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 743 . . . . . . 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 3168 . . . . . 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 4415 . . . . 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 2672 . . . 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 6107 . . . . 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 4593 . . . 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 743 . . 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 2610 . . . 4 if(𝐽 ∈ Top, 𝐽, {∅}) = if(𝐽 ∈ Top, 𝐽, {∅})
41 eqid 2610 . . . 4 (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42 eqid 2610 . . . 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 4760 . . . . . 6 ∅ ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
4443elimel 4100 . . . . 5 if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
45 elpwi 4117 . . . . 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 30451 . . 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 4090 . 2 ((𝐴𝑋𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14))
4948ancoms 468 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  cdif 3537  wss 3540  c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127   cuni 4372   cint 4410   class class class wbr 4583  cfv 5804  Fincfn 7841  1c1 9816  cle 9954  4c4 10949  cdc 11369  #chash 12979  Topctop 20517  clsccl 20632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-hash 12980  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635
This theorem is referenced by: (None)
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