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Mirrors > Home > HSE Home > Th. List > stcltr2i | Structured version Visualization version GIF version |
Description: Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stcltr1.1 | ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦))) |
stcltr1.2 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
stcltr2i | ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ ((𝑆‘𝐴) = 1 → ((𝑆‘ ℋ) = 1 → (𝑆‘𝐴) = 1)) | |
2 | stcltr1.1 | . . . 4 ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦))) | |
3 | helch 28689 | . . . 4 ⊢ ℋ ∈ Cℋ | |
4 | stcltr1.2 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
5 | 2, 3, 4 | stcltr1i 29722 | . . 3 ⊢ (𝜑 → (((𝑆‘ ℋ) = 1 → (𝑆‘𝐴) = 1) → ℋ ⊆ 𝐴)) |
6 | 1, 5 | syl5 34 | . 2 ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → ℋ ⊆ 𝐴)) |
7 | 4 | chssii 28677 | . . 3 ⊢ 𝐴 ⊆ ℋ |
8 | eqss 3836 | . . 3 ⊢ (𝐴 = ℋ ↔ (𝐴 ⊆ ℋ ∧ ℋ ⊆ 𝐴)) | |
9 | 7, 8 | mpbiran 699 | . 2 ⊢ (𝐴 = ℋ ↔ ℋ ⊆ 𝐴) |
10 | 6, 9 | syl6ibr 244 | 1 ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ⊆ wss 3792 ‘cfv 6137 1c1 10275 ℋchba 28365 Cℋ cch 28375 Statescst 28408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-1cn 10332 ax-addcl 10334 ax-hilex 28445 ax-hfvadd 28446 ax-hv0cl 28449 ax-hfvmul 28451 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-map 8144 df-nn 11380 df-hlim 28418 df-sh 28653 df-ch 28667 |
This theorem is referenced by: stcltrlem1 29724 |
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