chrelat2 - Hilbert Space Explorer

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Theorem chrelat2 29931

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 3884	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵))
2	1	notbid 310	. . 3 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (¬ 𝐴 ⊆ 𝐵 ↔ ¬ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ⊆ 𝐵))
3		sseq2 3885	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ)))
4	3	anbi1d 620	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ C_ℋ , 𝐴, ℋ) → ((𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∧ ¬ 𝑥 ⊆ 𝐵)))
5	4	rexbidv 3242	. . 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 338	. 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 3885	. . . 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 310	. . 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 3885	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
10	9	notbid 310	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ C_ℋ , 𝐵, ℋ) → (¬ 𝑥 ⊆ 𝐵 ↔ ¬ 𝑥 ⊆ if(𝐵 ∈ C_ℋ , 𝐵, ℋ)))
11	10	anbi2d 619	. . . 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 3242	. . 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 338	. 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 28803	. . 3 ⊢ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∈ C_ℋ
15		ifchhv 28803	. . 3 ⊢ if(𝐵 ∈ C_ℋ , 𝐵, ℋ) ∈ C_ℋ
16	14, 15	chrelat2i 29926	. 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 4408	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 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∃wrex 3089 ⊆ wss 3831 ifcif 4351 ℋchba 28478 C_ℋ cch 28488 HAtomscat 28524

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cc 9657 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 ax-hilex 28558 ax-hfvadd 28559 ax-hvcom 28560 ax-hvass 28561 ax-hv0cl 28562 ax-hvaddid 28563 ax-hfvmul 28564 ax-hvmulid 28565 ax-hvmulass 28566 ax-hvdistr1 28567 ax-hvdistr2 28568 ax-hvmul0 28569 ax-hfi 28638 ax-his1 28641 ax-his2 28642 ax-his3 28643 ax-his4 28644 ax-hcompl 28761

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-omul 7912 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-card 9164 df-acn 9167 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-clim 14709 df-rlim 14710 df-sum 14907 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-cn 21542 df-cnp 21543 df-lm 21544 df-haus 21630 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cfil 23564 df-cau 23565 df-cmet 23566 df-grpo 28050 df-gid 28051 df-ginv 28052 df-gdiv 28053 df-ablo 28102 df-vc 28116 df-nv 28149 df-va 28152 df-ba 28153 df-sm 28154 df-0v 28155 df-vs 28156 df-nmcv 28157 df-ims 28158 df-dip 28258 df-ssp 28279 df-ph 28370 df-cbn 28421 df-hnorm 28527 df-hba 28528 df-hvsub 28530 df-hlim 28531 df-hcau 28532 df-sh 28766 df-ch 28780 df-oc 28811 df-ch0 28812 df-shs 28869 df-span 28870 df-chj 28871 df-chsup 28872 df-cv 29840 df-at 29899

This theorem is referenced by: chrelat3 29932 mdsymlem3 29966

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