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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 28405 | . . . 4 ⊢ ℋ ∈ Cℋ | |
4 | stcltr1.2 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
5 | 2, 3, 4 | stcltr1i 29438 | . . 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 28393 | . . 3 ⊢ 𝐴 ⊆ ℋ |
8 | eqss 3755 | . . 3 ⊢ (𝐴 = ℋ ↔ (𝐴 ⊆ ℋ ∧ ℋ ⊆ 𝐴)) | |
9 | 7, 8 | mpbiran 991 | . 2 ⊢ (𝐴 = ℋ ↔ ℋ ⊆ 𝐴) |
10 | 6, 9 | syl6ibr 242 | 1 ⊢ (𝜑 → ((𝑆‘𝐴) = 1 → 𝐴 = ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1628 ∈ wcel 2135 ∀wral 3046 ⊆ wss 3711 ‘cfv 6045 1c1 10125 ℋchil 28081 Cℋ cch 28091 Statescst 28124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-i2m1 10192 ax-1ne0 10193 ax-rrecex 10196 ax-cnre 10197 ax-hilex 28161 ax-hfvadd 28162 ax-hv0cl 28165 ax-hfvmul 28167 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-ral 3051 df-rex 3052 df-reu 3053 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-om 7227 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-map 8021 df-nn 11209 df-hlim 28134 df-sh 28369 df-ch 28383 |
This theorem is referenced by: stcltrlem1 29440 |
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