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Mirrors > Home > HSE Home > Th. List > atss | Structured version Visualization version GIF version |
Description: A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atss | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elat2 32065 | . 2 ⊢ (𝐵 ∈ HAtoms ↔ (𝐵 ∈ Cℋ ∧ (𝐵 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → (𝑥 = 𝐵 ∨ 𝑥 = 0ℋ))))) | |
2 | sseq1 4000 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
3 | eqeq1 2728 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
4 | eqeq1 2728 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = 0ℋ ↔ 𝐴 = 0ℋ)) | |
5 | 3, 4 | orbi12d 915 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 ∨ 𝑥 = 0ℋ) ↔ (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ))) |
6 | 2, 5 | imbi12d 344 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐵 → (𝑥 = 𝐵 ∨ 𝑥 = 0ℋ)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ)))) |
7 | 6 | rspcv 3600 | . . . . 5 ⊢ (𝐴 ∈ Cℋ → (∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → (𝑥 = 𝐵 ∨ 𝑥 = 0ℋ)) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ)))) |
8 | 7 | adantld 490 | . . . 4 ⊢ (𝐴 ∈ Cℋ → ((𝐵 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → (𝑥 = 𝐵 ∨ 𝑥 = 0ℋ))) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ)))) |
9 | 8 | adantld 490 | . . 3 ⊢ (𝐴 ∈ Cℋ → ((𝐵 ∈ Cℋ ∧ (𝐵 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → (𝑥 = 𝐵 ∨ 𝑥 = 0ℋ)))) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ)))) |
10 | 9 | imp 406 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ (𝐵 ∈ Cℋ ∧ (𝐵 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐵 → (𝑥 = 𝐵 ∨ 𝑥 = 0ℋ))))) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ))) |
11 | 1, 10 | sylan2b 593 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = 0ℋ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3941 Cℋ cch 30654 0ℋc0h 30660 HAtomscat 30690 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30724 ax-hfvadd 30725 ax-hvcom 30726 ax-hvass 30727 ax-hv0cl 30728 ax-hvaddid 30729 ax-hfvmul 30730 ax-hvmulid 30731 ax-hvmulass 30732 ax-hvdistr1 30733 ax-hvdistr2 30734 ax-hvmul0 30735 ax-hfi 30804 ax-his1 30807 ax-his2 30808 ax-his3 30809 ax-his4 30810 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-n0 12471 df-z 12557 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-icc 13329 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-topgen 17390 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-top 22720 df-topon 22737 df-bases 22773 df-lm 23057 df-haus 23143 df-grpo 30218 df-gid 30219 df-ginv 30220 df-gdiv 30221 df-ablo 30270 df-vc 30284 df-nv 30317 df-va 30320 df-ba 30321 df-sm 30322 df-0v 30323 df-vs 30324 df-nmcv 30325 df-ims 30326 df-hnorm 30693 df-hvsub 30696 df-hlim 30697 df-sh 30932 df-ch 30946 df-ch0 30978 df-cv 32004 df-at 32063 |
This theorem is referenced by: atsseq 32072 |
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