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Mirrors > Home > HSE Home > Th. List > chintcl | Structured version Visualization version GIF version |
Description: The intersection (infimum) of a nonempty subset of Cℋ belongs to Cℋ. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chintcl | ⊢ ((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Cℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteq 4953 | . . 3 ⊢ (𝐴 = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → ∩ 𝐴 = ∩ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ )) | |
2 | 1 | eleq1d 2817 | . 2 ⊢ (𝐴 = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → (∩ 𝐴 ∈ Cℋ ↔ ∩ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ∈ Cℋ )) |
3 | sseq1 4007 | . . . . 5 ⊢ (𝐴 = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → (𝐴 ⊆ Cℋ ↔ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ⊆ Cℋ )) | |
4 | neeq1 3002 | . . . . 5 ⊢ (𝐴 = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → (𝐴 ≠ ∅ ↔ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ≠ ∅)) | |
5 | 3, 4 | anbi12d 630 | . . . 4 ⊢ (𝐴 = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → ((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) ↔ (if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ⊆ Cℋ ∧ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ≠ ∅))) |
6 | sseq1 4007 | . . . . 5 ⊢ ( Cℋ = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → ( Cℋ ⊆ Cℋ ↔ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ⊆ Cℋ )) | |
7 | neeq1 3002 | . . . . 5 ⊢ ( Cℋ = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → ( Cℋ ≠ ∅ ↔ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ≠ ∅)) | |
8 | 6, 7 | anbi12d 630 | . . . 4 ⊢ ( Cℋ = if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) → (( Cℋ ⊆ Cℋ ∧ Cℋ ≠ ∅) ↔ (if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ⊆ Cℋ ∧ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ≠ ∅))) |
9 | ssid 4004 | . . . . 5 ⊢ Cℋ ⊆ Cℋ | |
10 | h0elch 30790 | . . . . . 6 ⊢ 0ℋ ∈ Cℋ | |
11 | 10 | ne0ii 4337 | . . . . 5 ⊢ Cℋ ≠ ∅ |
12 | 9, 11 | pm3.2i 470 | . . . 4 ⊢ ( Cℋ ⊆ Cℋ ∧ Cℋ ≠ ∅) |
13 | 5, 8, 12 | elimhyp 4593 | . . 3 ⊢ (if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ⊆ Cℋ ∧ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ≠ ∅) |
14 | 13 | chintcli 30866 | . 2 ⊢ ∩ if((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅), 𝐴, Cℋ ) ∈ Cℋ |
15 | 2, 14 | dedth 4586 | 1 ⊢ ((𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Cℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ⊆ wss 3948 ∅c0 4322 ifcif 4528 ∩ cint 4950 Cℋ cch 30464 0ℋc0h 30470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 ax-hilex 30534 ax-hfvadd 30535 ax-hvcom 30536 ax-hvass 30537 ax-hv0cl 30538 ax-hvaddid 30539 ax-hfvmul 30540 ax-hvmulid 30541 ax-hvmulass 30542 ax-hvdistr1 30543 ax-hvdistr2 30544 ax-hvmul0 30545 ax-hfi 30614 ax-his1 30617 ax-his2 30618 ax-his3 30619 ax-his4 30620 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-icc 13338 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-topgen 17396 df-psmet 21140 df-xmet 21141 df-met 21142 df-bl 21143 df-mopn 21144 df-top 22629 df-topon 22646 df-bases 22682 df-lm 22966 df-haus 23052 df-grpo 30028 df-gid 30029 df-ginv 30030 df-gdiv 30031 df-ablo 30080 df-vc 30094 df-nv 30127 df-va 30130 df-ba 30131 df-sm 30132 df-0v 30133 df-vs 30134 df-nmcv 30135 df-ims 30136 df-hnorm 30503 df-hvsub 30506 df-hlim 30507 df-sh 30742 df-ch 30756 df-ch0 30788 |
This theorem is referenced by: ococin 30943 |
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