chrelat2i - Hilbert Space Explorer

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Theorem chrelat2i 32456

Description: A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
chpssat.1	⊢ 𝐴 ∈ C_ℋ
chpssat.2	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
chrelat2i	⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem chrelat2i**
Step	Hyp	Ref	Expression
1		nssinpss 4197	. . 3 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)
2		chpssat.1	. . . . . 6 ⊢ 𝐴 ∈ C_ℋ
3		chpssat.2	. . . . . 6 ⊢ 𝐵 ∈ C_ℋ
4	2, 3	chincli 31551	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
5	4, 2	chrelati 32455	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 → ∃𝑥 ∈ HAtoms ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴))
6		atelch 32435	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
7		chlub 31600	. . . . . . . . . 10 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴))
8	4, 2, 7	mp3an13 1461	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴))
9		simpr 486	. . . . . . . . 9 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
10	8, 9	biimtrrdi 256	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴 → 𝑥 ⊆ 𝐴))
11	10	adantld 492	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴) → 𝑥 ⊆ 𝐴))
12		ssin 4169	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
13	12	notbii 322	. . . . . . . . . 10 ⊢ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵))
14		chnle 31605	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
15	4, 14	mpan 697	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
16	13, 15	bitrid 285	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
17	16, 8	anbi12d 639	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → ((¬ (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 877	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
20		pm4.55 996	. . . . . . . . . . . 12 ⊢ (¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴))
21		imor 860	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
22	19, 20, 21	3bitr4ri 306	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ ¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴))
23	18, 22	sylib 220	. . . . . . . . . 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 723	. . . . . . . 8 ⊢ ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴)) → ¬ 𝑥 ⊆ 𝐵)
26	17, 25	biimtrrdi 256	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴) → ¬ 𝑥 ⊆ 𝐵))
27	11, 26	jcad 518	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (((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 3076	. . . 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 219	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
32		sstr2 3923	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
33	32	com12 32	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
34	33	ralrimivw 3137	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
35		iman 403	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
36	35	ralbii 3087	. . . . 5 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
37		ralnex 3067	. . . . 5 ⊢ (∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
38	36, 37	bitri 277	. . . 4 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
39	34, 38	sylib 220	. . 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 211	1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∨ wo 854 ∈ wcel 2121 ∀wral 3055 ∃wrex 3065 ∩ cin 3883 ⊆ wss 3884 ⊊ wpss 3885 (class class class)co 7359 C_ℋ cch 31020 ∨_ℋ chj 31024 HAtomscat 31056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-inf2 9557 ax-cc 10353 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112 ax-addf 11113 ax-mulf 11114 ax-hilex 31090 ax-hfvadd 31091 ax-hvcom 31092 ax-hvass 31093 ax-hv0cl 31094 ax-hvaddid 31095 ax-hfvmul 31096 ax-hvmulid 31097 ax-hvmulass 31098 ax-hvdistr1 31099 ax-hvdistr2 31100 ax-hvmul0 31101 ax-hfi 31170 ax-his1 31173 ax-his2 31174 ax-his3 31175 ax-his4 31176 ax-hcompl 31293

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-acn 9861 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sum 15644 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19286 df-cmn 19751 df-psmet 21342 df-xmet 21343 df-met 21344 df-bl 21345 df-mopn 21346 df-fbas 21347 df-fg 21348 df-cnfld 21351 df-top 22880 df-topon 22897 df-topsp 22919 df-bases 22932 df-cld 23005 df-ntr 23006 df-cls 23007 df-nei 23084 df-cn 23213 df-cnp 23214 df-lm 23215 df-haus 23301 df-tx 23548 df-hmeo 23741 df-fil 23832 df-fm 23924 df-flim 23925 df-flf 23926 df-xms 24306 df-ms 24307 df-tms 24308 df-cfil 25243 df-cau 25244 df-cmet 25245 df-grpo 30584 df-gid 30585 df-ginv 30586 df-gdiv 30587 df-ablo 30636 df-vc 30650 df-nv 30683 df-va 30686 df-ba 30687 df-sm 30688 df-0v 30689 df-vs 30690 df-nmcv 30691 df-ims 30692 df-dip 30792 df-ssp 30813 df-ph 30904 df-cbn 30954 df-hnorm 31059 df-hba 31060 df-hvsub 31062 df-hlim 31063 df-hcau 31064 df-sh 31298 df-ch 31312 df-oc 31343 df-ch0 31344 df-shs 31399 df-span 31400 df-chj 31401 df-chsup 31402 df-cv 32370 df-at 32429

This theorem is referenced by: chrelat2 32461

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