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Mirrors > Home > HSE Home > Th. List > chrelat2i | Structured version Visualization version GIF version |
Description: A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chpssat.1 | ⊢ 𝐴 ∈ Cℋ |
chpssat.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chrelat2i | ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nssinpss 4232 | . . 3 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) | |
2 | chpssat.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
3 | chpssat.2 | . . . . . 6 ⊢ 𝐵 ∈ Cℋ | |
4 | 2, 3 | chincli 29165 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
5 | 4, 2 | chrelati 30069 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 → ∃𝑥 ∈ HAtoms ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴)) |
6 | atelch 30049 | . . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ Cℋ ) | |
7 | chlub 29214 | . . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴)) | |
8 | 4, 2, 7 | mp3an13 1443 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴)) |
9 | simpr 485 | . . . . . . . . 9 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐴) | |
10 | 8, 9 | syl6bir 255 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)) |
11 | 10 | adantld 491 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)) |
12 | ssin 4206 | . . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
13 | 12 | notbii 321 | . . . . . . . . . 10 ⊢ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) |
14 | chnle 29219 | . . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥))) | |
15 | 4, 14 | mpan 686 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥))) |
16 | 13, 15 | syl5bb 284 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥))) |
17 | 16, 8 | anbi12d 630 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴)) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴))) |
18 | pm3.21 472 | . . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) | |
19 | orcom 864 | . . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) | |
20 | pm4.55 981 | . . . . . . . . . . . 12 ⊢ (¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴)) | |
21 | imor 847 | . . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) | |
22 | 19, 20, 21 | 3bitr4ri 305 | . . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ ¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴)) |
23 | 18, 22 | sylib 219 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → ¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴)) |
24 | 23 | con2i 141 | . . . . . . . . 9 ⊢ ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴) → ¬ 𝑥 ⊆ 𝐵) |
25 | 24 | adantrl 712 | . . . . . . . 8 ⊢ ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴)) → ¬ 𝑥 ⊆ 𝐵) |
26 | 17, 25 | syl6bir 255 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → ¬ 𝑥 ⊆ 𝐵)) |
27 | 11, 26 | jcad 513 | . . . . . 6 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))) |
28 | 6, 27 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ HAtoms → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))) |
29 | 28 | reximia 3242 | . . . 4 ⊢ (∃𝑥 ∈ HAtoms ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
30 | 5, 29 | syl 17 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
31 | 1, 30 | sylbi 218 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
32 | sstr2 3973 | . . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵)) | |
33 | 32 | com12 32 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) |
34 | 33 | ralrimivw 3183 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) |
35 | iman 402 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) | |
36 | 35 | ralbii 3165 | . . . . 5 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
37 | ralnex 3236 | . . . . 5 ⊢ (∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) | |
38 | 36, 37 | bitri 276 | . . . 4 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
39 | 34, 38 | sylib 219 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
40 | 39 | con2i 141 | . 2 ⊢ (∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵) |
41 | 31, 40 | impbii 210 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∈ wcel 2105 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 ⊆ wss 3935 ⊊ wpss 3936 (class class class)co 7145 Cℋ cch 28634 ∨ℋ chj 28638 HAtomscat 28670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28704 ax-hfvadd 28705 ax-hvcom 28706 ax-hvass 28707 ax-hv0cl 28708 ax-hvaddid 28709 ax-hfvmul 28710 ax-hvmulid 28711 ax-hvmulass 28712 ax-hvdistr1 28713 ax-hvdistr2 28714 ax-hvmul0 28715 ax-hfi 28784 ax-his1 28787 ax-his2 28788 ax-his3 28789 ax-his4 28790 ax-hcompl 28907 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 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 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 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-se 5509 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-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-omul 8098 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-acn 9360 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-rlim 14836 df-sum 15033 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-cn 21765 df-cnp 21766 df-lm 21767 df-haus 21853 df-tx 22100 df-hmeo 22293 df-fil 22384 df-fm 22476 df-flim 22477 df-flf 22478 df-xms 22859 df-ms 22860 df-tms 22861 df-cfil 23787 df-cau 23788 df-cmet 23789 df-grpo 28198 df-gid 28199 df-ginv 28200 df-gdiv 28201 df-ablo 28250 df-vc 28264 df-nv 28297 df-va 28300 df-ba 28301 df-sm 28302 df-0v 28303 df-vs 28304 df-nmcv 28305 df-ims 28306 df-dip 28406 df-ssp 28427 df-ph 28518 df-cbn 28568 df-hnorm 28673 df-hba 28674 df-hvsub 28676 df-hlim 28677 df-hcau 28678 df-sh 28912 df-ch 28926 df-oc 28957 df-ch0 28958 df-shs 29013 df-span 29014 df-chj 29015 df-chsup 29016 df-cv 29984 df-at 30043 |
This theorem is referenced by: chrelat2 30075 |
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