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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 28100 | . . . 4 ⊢ ℋ ∈ Cℋ | |
4 | stcltr1.2 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
5 | 2, 3, 4 | stcltr1i 29133 | . . 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 28088 | . . 3 ⊢ 𝐴 ⊆ ℋ |
8 | eqss 3618 | . . 3 ⊢ (𝐴 = ℋ ↔ (𝐴 ⊆ ℋ ∧ ℋ ⊆ 𝐴)) | |
9 | 7, 8 | mpbiran 953 | . 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 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ‘cfv 5888 1c1 9937 ℋchil 27776 Cℋ cch 27786 Statescst 27819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 ax-hilex 27856 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-map 7859 df-nn 11021 df-hlim 27829 df-sh 28064 df-ch 28078 |
This theorem is referenced by: stcltrlem1 29135 |
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