![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem10 | Structured version Visualization version GIF version |
Description: Lemma for kur14 35200. Discharge the set 𝑇. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem10.j | ⊢ 𝐽 ∈ Top |
kur14lem10.x | ⊢ 𝑋 = ∪ 𝐽 |
kur14lem10.k | ⊢ 𝐾 = (cls‘𝐽) |
kur14lem10.s | ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} |
kur14lem10.a | ⊢ 𝐴 ⊆ 𝑋 |
Ref | Expression |
---|---|
kur14lem10 | ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem10.j | . 2 ⊢ 𝐽 ∈ Top | |
2 | kur14lem10.x | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
3 | kur14lem10.k | . 2 ⊢ 𝐾 = (cls‘𝐽) | |
4 | eqid 2734 | . 2 ⊢ (int‘𝐽) = (int‘𝐽) | |
5 | kur14lem10.a | . 2 ⊢ 𝐴 ⊆ 𝑋 | |
6 | eqid 2734 | . 2 ⊢ (𝑋 ∖ (𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘𝐴)) | |
7 | eqid 2734 | . 2 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = (𝐾‘(𝑋 ∖ 𝐴)) | |
8 | eqid 2734 | . 2 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴)) | |
9 | eqid 2734 | . 2 ⊢ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {(𝑋 ∖ (𝐾‘𝐴)), (𝐾‘(𝑋 ∖ 𝐴)), ((int‘𝐽)‘𝐴)}) ∪ {(𝐾‘(𝑋 ∖ (𝐾‘𝐴))), ((int‘𝐽)‘(𝐾‘𝐴)), (𝐾‘((int‘𝐽)‘𝐴))}) ∪ ({((int‘𝐽)‘(𝐾‘(𝑋 ∖ 𝐴))), (𝐾‘((int‘𝐽)‘(𝐾‘𝐴))), ((int‘𝐽)‘(𝐾‘(𝑋 ∖ (𝐾‘𝐴))))} ∪ {(𝐾‘((int‘𝐽)‘(𝐾‘(𝑋 ∖ 𝐴)))), ((int‘𝐽)‘(𝐾‘((int‘𝐽)‘𝐴)))})) = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {(𝑋 ∖ (𝐾‘𝐴)), (𝐾‘(𝑋 ∖ 𝐴)), ((int‘𝐽)‘𝐴)}) ∪ {(𝐾‘(𝑋 ∖ (𝐾‘𝐴))), ((int‘𝐽)‘(𝐾‘𝐴)), (𝐾‘((int‘𝐽)‘𝐴))}) ∪ ({((int‘𝐽)‘(𝐾‘(𝑋 ∖ 𝐴))), (𝐾‘((int‘𝐽)‘(𝐾‘𝐴))), ((int‘𝐽)‘(𝐾‘(𝑋 ∖ (𝐾‘𝐴))))} ∪ {(𝐾‘((int‘𝐽)‘(𝐾‘(𝑋 ∖ 𝐴)))), ((int‘𝐽)‘(𝐾‘((int‘𝐽)‘𝐴)))})) | |
10 | kur14lem10.s | . 2 ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | kur14lem9 35198 | 1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 {crab 3432 ∖ cdif 3959 ∪ cun 3960 ⊆ wss 3962 𝒫 cpw 4604 {cpr 4632 {ctp 4634 ∪ cuni 4911 ∩ cint 4950 class class class wbr 5147 ‘cfv 6562 Fincfn 8983 1c1 11153 ≤ cle 11293 4c4 12320 ;cdc 12730 ♯chash 14365 Topctop 22914 intcnt 23040 clsccl 23041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-hash 14366 df-top 22915 df-cld 23042 df-ntr 23043 df-cls 23044 |
This theorem is referenced by: kur14 35200 |
Copyright terms: Public domain | W3C validator |