chrelat2 - Hilbert Space Explorer

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Theorem chrelat2 30145

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 3985	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵))
2	1	notbid 320	. . 3 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (¬ 𝐴 ⊆ 𝐵 ↔ ¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵))
3		sseq2 3986	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ)))
4	3	anbi1d 631	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → ((𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵)))
5	4	rexbidv 3296	. . 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 348	. 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 3986	. . . 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 320	. . 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 3986	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
10	9	notbid 320	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (¬ 𝑥 ⊆ 𝐵 ↔ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
11	10	anbi2d 630	. . . 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 3296	. . 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 348	. 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 29019	. . 3 ⊢ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∈ C_ℋ
15		ifchhv 29019	. . 3 ⊢ if(𝐵 ∈ C_ℋ , 𝐵, ℋ) ∈ C_ℋ
16	14, 15	chrelat2i 30140	. 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 4517	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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ⊆ wss 3929 ifcif 4460 ℋchba 28694 C_ℋ cch 28704 HAtomscat 28740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cc 9850 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 ax-hilex 28774 ax-hfvadd 28775 ax-hvcom 28776 ax-hvass 28777 ax-hv0cl 28778 ax-hvaddid 28779 ax-hfvmul 28780 ax-hvmulid 28781 ax-hvmulass 28782 ax-hvdistr1 28783 ax-hvdistr2 28784 ax-hvmul0 28785 ax-hfi 28854 ax-his1 28857 ax-his2 28858 ax-his3 28859 ax-his4 28860 ax-hcompl 28977

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-omul 8100 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-fi 8868 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-acn 9364 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12890 df-fzo 13031 df-fl 13159 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-rlim 14841 df-sum 15038 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-rest 16691 df-topn 16692 df-0g 16710 df-gsum 16711 df-topgen 16712 df-pt 16713 df-prds 16716 df-xrs 16770 df-qtop 16775 df-imas 16776 df-xps 16778 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-mulg 18220 df-cntz 18442 df-cmn 18903 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-fbas 20537 df-fg 20538 df-cnfld 20541 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-cld 21622 df-ntr 21623 df-cls 21624 df-nei 21701 df-cn 21830 df-cnp 21831 df-lm 21832 df-haus 21918 df-tx 22165 df-hmeo 22358 df-fil 22449 df-fm 22541 df-flim 22542 df-flf 22543 df-xms 22925 df-ms 22926 df-tms 22927 df-cfil 23853 df-cau 23854 df-cmet 23855 df-grpo 28268 df-gid 28269 df-ginv 28270 df-gdiv 28271 df-ablo 28320 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-vs 28374 df-nmcv 28375 df-ims 28376 df-dip 28476 df-ssp 28497 df-ph 28588 df-cbn 28638 df-hnorm 28743 df-hba 28744 df-hvsub 28746 df-hlim 28747 df-hcau 28748 df-sh 28982 df-ch 28996 df-oc 29027 df-ch0 29028 df-shs 29083 df-span 29084 df-chj 29085 df-chsup 29086 df-cv 30054 df-at 30113

This theorem is referenced by: chrelat3 30146 mdsymlem3 30180

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