Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  kur14 Structured version   Visualization version   GIF version

Theorem kur14 31324
 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 2718 . . . . . . . . 9 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝐴𝑥 ↔ if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥))
32anbi1d 741 . . . . . . . 8 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)))
43rabbidv 3220 . . . . . . 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 4512 . . . . . 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 2697 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)})
76eleq1d 2715 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ∈ Fin))
86fveq2d 6233 . . . . 5 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → (#‘𝑆) = (#‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}))
98breq1d 4695 . . . 4 (𝐴 = if(𝐴𝑋, 𝐴, ∅) → ((#‘𝑆) ≤ 14 ↔ (#‘ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}) ≤ 14))
107, 9anbi12d 747 . . 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 4476 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}))
1311, 12syl5eq 2697 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = if(𝐽 ∈ Top, 𝐽, {∅}))
1413pweqd 4196 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1514pweqd 4196 . . . . . . 7 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}))
1613sseq2d 3666 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 if(𝐽 ∈ Top, 𝐽, {∅})))
17 sn0top 20851 . . . . . . . . . . . . . 14 {∅} ∈ Top
1817elimel 4183 . . . . . . . . . . . . 13 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19 uniexg 6997 . . . . . . . . . . . . 13 (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
2018, 19ax-mp 5 . . . . . . . . . . . 12 if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
2120elpw2 4858 . . . . . . . . . . 11 (𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 if(𝐽 ∈ Top, 𝐽, {∅}))
2216, 21syl6bbr 278 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴𝑋𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})))
2322ifbid 4141 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
2423eleq1d 2715 . . . . . . . 8 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
2513difeq1d 3760 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋𝑦) = ( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26 kur14.k . . . . . . . . . . . . 13 𝐾 = (cls‘𝐽)
27 fveq2 6229 . . . . . . . . . . . . 13 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2826, 27syl5eq 2697 . . . . . . . . . . . 12 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
2928fveq1d 6231 . . . . . . . . . . 11 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
3025, 29preq12d 4308 . . . . . . . . . 10 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋𝑦), (𝐾𝑦)} = {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
3130sseq1d 3665 . . . . . . . . 9 (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {( if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
3231ralbidv 3015 . . . . . . . 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 747 . . . . . . 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 3226 . . . . . 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 4512 . . . . 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 2715 . . . 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 6233 . . . . 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 4695 . . . 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 747 . . 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 2651 . . . 4 if(𝐽 ∈ Top, 𝐽, {∅}) = if(𝐽 ∈ Top, 𝐽, {∅})
41 eqid 2651 . . . 4 (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42 eqid 2651 . . . 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 4864 . . . . . 6 ∅ ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
4443elimel 4183 . . . . 5 if(𝐴 ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 if(𝐽 ∈ Top, 𝐽, {∅})
45 elpwi 4201 . . . . 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 31323 . . 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 4173 . 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 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210  {cpr 4212  ∪ cuni 4468  ∩ cint 4507   class class class wbr 4685  ‘cfv 5926  Fincfn 7997  1c1 9975   ≤ cle 10113  4c4 11110  ;cdc 11531  #chash 13157  Topctop 20746  clsccl 20870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-hash 13158  df-top 20747  df-topon 20764  df-cld 20871  df-ntr 20872  df-cls 20873 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator