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 32458. 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 2821 | . 2 ⊢ (int‘𝐽) = (int‘𝐽) | |
5 | kur14lem10.a | . 2 ⊢ 𝐴 ⊆ 𝑋 | |
6 | eqid 2821 | . 2 ⊢ (𝑋 ∖ (𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘𝐴)) | |
7 | eqid 2821 | . 2 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = (𝐾‘(𝑋 ∖ 𝐴)) | |
8 | eqid 2821 | . 2 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴)) | |
9 | eqid 2821 | . 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 32456 | 1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 𝒫 cpw 4538 {cpr 4562 {ctp 4564 ∪ cuni 4831 ∩ cint 4868 class class class wbr 5058 ‘cfv 6349 Fincfn 8503 1c1 10532 ≤ cle 10670 4c4 11688 ;cdc 12092 ♯chash 13684 Topctop 21495 intcnt 21619 clsccl 21620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-hash 13685 df-top 21496 df-cld 21621 df-ntr 21622 df-cls 21623 |
This theorem is referenced by: kur14 32458 |
Copyright terms: Public domain | W3C validator |