![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 4215 | . . 3 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) | |
2 | chpssat.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
3 | chpssat.2 | . . . . . 6 ⊢ 𝐵 ∈ Cℋ | |
4 | 2, 3 | chincli 30231 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
5 | 4, 2 | chrelati 31135 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 → ∃𝑥 ∈ HAtoms ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴)) |
6 | atelch 31115 | . . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ Cℋ ) | |
7 | chlub 30280 | . . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴)) | |
8 | 4, 2, 7 | mp3an13 1453 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴)) |
9 | simpr 486 | . . . . . . . . 9 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐴) | |
10 | 8, 9 | syl6bir 254 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴 → 𝑥 ⊆ 𝐴)) |
11 | 10 | adantld 492 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)) |
12 | ssin 4189 | . . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
13 | 12 | notbii 320 | . . . . . . . . . 10 ⊢ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) |
14 | chnle 30285 | . . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥))) | |
15 | 4, 14 | mpan 689 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥))) |
16 | 13, 15 | bitrid 283 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥))) |
17 | 16, 8 | anbi12d 632 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴)) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴))) |
18 | pm3.21 473 | . . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) | |
19 | orcom 869 | . . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) | |
20 | pm4.55 987 | . . . . . . . . . . . 12 ⊢ (¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴)) | |
21 | imor 852 | . . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) | |
22 | 19, 20, 21 | 3bitr4ri 304 | . . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ ¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴)) |
23 | 18, 22 | sylib 217 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → ¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴)) |
24 | 23 | con2i 139 | . . . . . . . . 9 ⊢ ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴) → ¬ 𝑥 ⊆ 𝐵) |
25 | 24 | adantrl 715 | . . . . . . . 8 ⊢ ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴)) → ¬ 𝑥 ⊆ 𝐵) |
26 | 17, 25 | syl6bir 254 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨ℋ 𝑥) ⊆ 𝐴) → ¬ 𝑥 ⊆ 𝐵)) |
27 | 11, 26 | jcad 514 | . . . . . 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 3083 | . . . 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 216 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
32 | sstr2 3950 | . . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵)) | |
33 | 32 | com12 32 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) |
34 | 33 | ralrimivw 3146 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) |
35 | iman 403 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) | |
36 | 35 | ralbii 3095 | . . . . 5 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
37 | ralnex 3074 | . . . . 5 ⊢ (∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) | |
38 | 36, 37 | bitri 275 | . . . 4 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
39 | 34, 38 | sylib 217 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
40 | 39 | con2i 139 | . 2 ⊢ (∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵) |
41 | 31, 40 | impbii 208 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∈ wcel 2107 ∀wral 3063 ∃wrex 3072 ∩ cin 3908 ⊆ wss 3909 ⊊ wpss 3910 (class class class)co 7352 Cℋ cch 29700 ∨ℋ chj 29704 HAtomscat 29736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-inf2 9536 ax-cc 10330 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 ax-addf 11089 ax-mulf 11090 ax-hilex 29770 ax-hfvadd 29771 ax-hvcom 29772 ax-hvass 29773 ax-hv0cl 29774 ax-hvaddid 29775 ax-hfvmul 29776 ax-hvmulid 29777 ax-hvmulass 29778 ax-hvdistr1 29779 ax-hvdistr2 29780 ax-hvmul0 29781 ax-hfi 29850 ax-his1 29853 ax-his2 29854 ax-his3 29855 ax-his4 29856 ax-hcompl 29973 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-1st 7914 df-2nd 7915 df-supp 8086 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8607 df-map 8726 df-pm 8727 df-ixp 8795 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fsupp 9265 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9405 df-card 9834 df-acn 9837 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-q 12829 df-rp 12871 df-xneg 12988 df-xadd 12989 df-xmul 12990 df-ioo 13223 df-ico 13225 df-icc 13226 df-fz 13380 df-fzo 13523 df-fl 13652 df-seq 13862 df-exp 13923 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-rlim 15331 df-sum 15531 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-hom 17117 df-cco 17118 df-rest 17264 df-topn 17265 df-0g 17283 df-gsum 17284 df-topgen 17285 df-pt 17286 df-prds 17289 df-xrs 17344 df-qtop 17349 df-imas 17350 df-xps 17352 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-mulg 18832 df-cntz 19056 df-cmn 19523 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-fbas 20746 df-fg 20747 df-cnfld 20750 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-cld 22322 df-ntr 22323 df-cls 22324 df-nei 22401 df-cn 22530 df-cnp 22531 df-lm 22532 df-haus 22618 df-tx 22865 df-hmeo 23058 df-fil 23149 df-fm 23241 df-flim 23242 df-flf 23243 df-xms 23625 df-ms 23626 df-tms 23627 df-cfil 24571 df-cau 24572 df-cmet 24573 df-grpo 29264 df-gid 29265 df-ginv 29266 df-gdiv 29267 df-ablo 29316 df-vc 29330 df-nv 29363 df-va 29366 df-ba 29367 df-sm 29368 df-0v 29369 df-vs 29370 df-nmcv 29371 df-ims 29372 df-dip 29472 df-ssp 29493 df-ph 29584 df-cbn 29634 df-hnorm 29739 df-hba 29740 df-hvsub 29742 df-hlim 29743 df-hcau 29744 df-sh 29978 df-ch 29992 df-oc 30023 df-ch0 30024 df-shs 30079 df-span 30080 df-chj 30081 df-chsup 30082 df-cv 31050 df-at 31109 |
This theorem is referenced by: chrelat2 31141 |
Copyright terms: Public domain | W3C validator |