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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem10 | Structured version Visualization version GIF version |
Description: Lemma for kur14 31797. 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 2777 | . 2 ⊢ (int‘𝐽) = (int‘𝐽) | |
5 | kur14lem10.a | . 2 ⊢ 𝐴 ⊆ 𝑋 | |
6 | eqid 2777 | . 2 ⊢ (𝑋 ∖ (𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘𝐴)) | |
7 | eqid 2777 | . 2 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = (𝐾‘(𝑋 ∖ 𝐴)) | |
8 | eqid 2777 | . 2 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴)) | |
9 | eqid 2777 | . 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 31795 | 1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 {crab 3093 ∖ cdif 3788 ∪ cun 3789 ⊆ wss 3791 𝒫 cpw 4378 {cpr 4399 {ctp 4401 ∪ cuni 4671 ∩ cint 4710 class class class wbr 4886 ‘cfv 6135 Fincfn 8241 1c1 10273 ≤ cle 10412 4c4 11432 ;cdc 11845 ♯chash 13435 Topctop 21105 intcnt 21229 clsccl 21230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-hash 13436 df-top 21106 df-cld 21231 df-ntr 21232 df-cls 21233 |
This theorem is referenced by: kur14 31797 |
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