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Mirrors > Home > HSE Home > Th. List > shunssji | Structured version Visualization version GIF version |
Description: Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shincl.1 | ⊢ 𝐴 ∈ Sℋ |
shincl.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shunssji | ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shincl.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
2 | 1 | shssii 28992 | . . . 4 ⊢ 𝐴 ⊆ ℋ |
3 | shincl.2 | . . . . 5 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | shssii 28992 | . . . 4 ⊢ 𝐵 ⊆ ℋ |
5 | 2, 4 | unssi 4163 | . . 3 ⊢ (𝐴 ∪ 𝐵) ⊆ ℋ |
6 | ococss 29072 | . . 3 ⊢ ((𝐴 ∪ 𝐵) ⊆ ℋ → (𝐴 ∪ 𝐵) ⊆ (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (𝐴 ∪ 𝐵) ⊆ (⊥‘(⊥‘(𝐴 ∪ 𝐵))) |
8 | shjval 29130 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | |
9 | 1, 3, 8 | mp2an 690 | . 2 ⊢ (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))) |
10 | 7, 9 | sseqtrri 4006 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∨ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 ℋchba 28698 Sℋ csh 28707 ⊥cort 28709 ∨ℋ chj 28712 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hilex 28778 ax-hfvadd 28779 ax-hv0cl 28782 ax-hfvmul 28784 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-sh 28986 df-oc 29031 df-chj 29089 |
This theorem is referenced by: shsleji 29149 chunssji 29246 |
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