chrelat2 - Hilbert Space Explorer

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Theorem chrelat2 32047

Description: A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)

**Assertion**
Ref	Expression
chrelat2	⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem chrelat2**
Step	Hyp	Ref	Expression
1		sseq1 3999	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵))
2	1	notbid 318	. . 3 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (¬ 𝐴 ⊆ 𝐵 ↔ ¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵))
3		sseq2 4000	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ)))
4	3	anbi1d 629	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → ((𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵)))
5	4	rexbidv 3170	. . 3 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵)))
6	2, 5	bibi12d 345	. 2 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → ((¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) ↔ (¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵))))
7		sseq2 4000	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵 ↔ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
8	7	notbid 318	. . 3 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵 ↔ ¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
9		sseq2 4000	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
10	9	notbid 318	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (¬ 𝑥 ⊆ 𝐵 ↔ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
11	10	anbi2d 628	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → ((𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ))))
12	11	rexbidv 3170	. . 3 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ))))
13	8, 12	bibi12d 345	. 2 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → ((¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵)) ↔ (¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ) ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))))
14		ifchhv 30921	. . 3 ⊢ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∈ C_ℋ
15		ifchhv 30921	. . 3 ⊢ if(𝐵 ∈ C_ℋ , 𝐵, ℋ) ∈ C_ℋ
16	14, 15	chrelat2i 32042	. 2 ⊢ (¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ) ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
17	6, 13, 16	dedth2h 4579	1 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 ⊆ wss 3940 ifcif 4520 ℋchba 30596 C_ℋ cch 30606 HAtomscat 30642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 ax-hilex 30676 ax-hfvadd 30677 ax-hvcom 30678 ax-hvass 30679 ax-hv0cl 30680 ax-hvaddid 30681 ax-hfvmul 30682 ax-hvmulid 30683 ax-hvmulass 30684 ax-hvdistr1 30685 ax-hvdistr2 30686 ax-hvmul0 30687 ax-hfi 30756 ax-his1 30759 ax-his2 30760 ax-his3 30761 ax-his4 30762 ax-hcompl 30879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-acn 9932 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-cn 23041 df-cnp 23042 df-lm 23043 df-haus 23129 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cfil 25093 df-cau 25094 df-cmet 25095 df-grpo 30170 df-gid 30171 df-ginv 30172 df-gdiv 30173 df-ablo 30222 df-vc 30236 df-nv 30269 df-va 30272 df-ba 30273 df-sm 30274 df-0v 30275 df-vs 30276 df-nmcv 30277 df-ims 30278 df-dip 30378 df-ssp 30399 df-ph 30490 df-cbn 30540 df-hnorm 30645 df-hba 30646 df-hvsub 30648 df-hlim 30649 df-hcau 30650 df-sh 30884 df-ch 30898 df-oc 30929 df-ch0 30930 df-shs 30985 df-span 30986 df-chj 30987 df-chsup 30988 df-cv 31956 df-at 32015

This theorem is referenced by: chrelat3 32048 mdsymlem3 32082

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