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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 29022 | . . . 4 ⊢ ℋ ∈ Cℋ | |
4 | stcltr1.2 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
5 | 2, 3, 4 | stcltr1i 30053 | . . 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 29010 | . . 3 ⊢ 𝐴 ⊆ ℋ |
8 | eqss 3984 | . . 3 ⊢ (𝐴 = ℋ ↔ (𝐴 ⊆ ℋ ∧ ℋ ⊆ 𝐴)) | |
9 | 7, 8 | mpbiran 707 | . 2 ⊢ (𝐴 = ℋ ↔ ℋ ⊆ 𝐴) |
10 | 6, 9 | syl6ibr 254 | 1 ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 ‘cfv 6357 1c1 10540 ℋchba 28698 Cℋ cch 28708 Statescst 28741 |
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-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 ax-hilex 28778 ax-hfvadd 28779 ax-hv0cl 28782 ax-hfvmul 28784 |
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-ral 3145 df-rex 3146 df-reu 3147 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-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 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-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-map 8410 df-nn 11641 df-hlim 28751 df-sh 28986 df-ch 29000 |
This theorem is referenced by: stcltrlem1 30055 |
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