chrelat2i - Hilbert Space Explorer

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Theorem chrelat2i 30446

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 4171	. . 3 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴)
2		chpssat.1	. . . . . 6 ⊢ 𝐴 ∈ C_ℋ
3		chpssat.2	. . . . . 6 ⊢ 𝐵 ∈ C_ℋ
4	2, 3	chincli 29541	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
5	4, 2	chrelati 30445	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 → ∃𝑥 ∈ HAtoms ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴))
6		atelch 30425	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
7		chlub 29590	. . . . . . . . . 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 1454	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴))
9		simpr 488	. . . . . . . . 9 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
10	8, 9	syl6bir 257	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴 → 𝑥 ⊆ 𝐴))
11	10	adantld 494	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴) → 𝑥 ⊆ 𝐴))
12		ssin 4145	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
13	12	notbii 323	. . . . . . . . . 10 ⊢ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵))
14		chnle 29595	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
15	4, 14	mpan 690	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
16	13, 15	syl5bb 286	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
17	16, 8	anbi12d 634	. . . . . . . 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		pm3.21 475	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
19		orcom 870	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
20		pm4.55 988	. . . . . . . . . . . 12 ⊢ (¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∨ ¬ 𝑥 ⊆ 𝐴))
21		imor 853	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ (¬ 𝑥 ⊆ 𝐴 ∨ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
22	19, 20, 21	3bitr4ri 307	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) ↔ ¬ (¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑥 ⊆ 𝐴))
23	18, 22	sylib 221	. . . . . . . . . 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 716	. . . . . . . 8 ⊢ ((¬ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ∧ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴)) → ¬ 𝑥 ⊆ 𝐵)
26	17, 25	syl6bir 257	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐴) → ¬ 𝑥 ⊆ 𝐵))
27	11, 26	jcad 516	. . . . . 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 3165	. . . 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 220	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
32		sstr2 3908	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
33	32	com12 32	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
34	33	ralrimivw 3106	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
35		iman 405	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
36	35	ralbii 3088	. . . . 5 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
37		ralnex 3158	. . . . 5 ⊢ (∀𝑥 ∈ HAtoms ¬ (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
38	36, 37	bitri 278	. . . 4 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵) ↔ ¬ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
39	34, 38	sylib 221	. . 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 212	1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 ∩ cin 3865 ⊆ wss 3866 ⊊ wpss 3867 (class class class)co 7213 C_ℋ cch 29010 ∨_ℋ chj 29014 HAtomscat 29046

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 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 ax-hilex 29080 ax-hfvadd 29081 ax-hvcom 29082 ax-hvass 29083 ax-hv0cl 29084 ax-hvaddid 29085 ax-hfvmul 29086 ax-hvmulid 29087 ax-hvmulass 29088 ax-hvdistr1 29089 ax-hvdistr2 29090 ax-hvmul0 29091 ax-hfi 29160 ax-his1 29163 ax-his2 29164 ax-his3 29165 ax-his4 29166 ax-hcompl 29283

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-cn 22124 df-cnp 22125 df-lm 22126 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cfil 24152 df-cau 24153 df-cmet 24154 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 df-dip 28782 df-ssp 28803 df-ph 28894 df-cbn 28944 df-hnorm 29049 df-hba 29050 df-hvsub 29052 df-hlim 29053 df-hcau 29054 df-sh 29288 df-ch 29302 df-oc 29333 df-ch0 29334 df-shs 29389 df-span 29390 df-chj 29391 df-chsup 29392 df-cv 30360 df-at 30419

This theorem is referenced by: chrelat2 30451

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