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Mirrors > Home > HSE Home > Th. List > stcltr1i | 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 |
stcltr1.3 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
stcltr1i | ⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stcltr1.1 | . 2 ⊢ (𝜑 ↔ (𝑆 ∈ States ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦))) | |
2 | stcltr1.2 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
3 | stcltr1.3 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
4 | fveqeq2 6505 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) = 1 ↔ (𝑆‘𝐴) = 1)) | |
5 | 4 | imbi1d 334 | . . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) ↔ ((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1))) |
6 | sseq1 3876 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦)) | |
7 | 5, 6 | imbi12d 337 | . . . 4 ⊢ (𝑥 = 𝐴 → ((((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) ↔ (((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) → 𝐴 ⊆ 𝑦))) |
8 | fveqeq2 6505 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝑦) = 1 ↔ (𝑆‘𝐵) = 1)) | |
9 | 8 | imbi2d 333 | . . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) ↔ ((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1))) |
10 | sseq2 3877 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵)) | |
11 | 9, 10 | imbi12d 337 | . . . 4 ⊢ (𝑦 = 𝐵 → ((((𝑆‘𝐴) = 1 → (𝑆‘𝑦) = 1) → 𝐴 ⊆ 𝑦) ↔ (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))) |
12 | 7, 11 | rspc2v 3542 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵))) |
13 | 2, 3, 12 | mp2an 679 | . 2 ⊢ (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (((𝑆‘𝑥) = 1 → (𝑆‘𝑦) = 1) → 𝑥 ⊆ 𝑦) → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)) |
14 | 1, 13 | simplbiim 497 | 1 ⊢ (𝜑 → (((𝑆‘𝐴) = 1 → (𝑆‘𝐵) = 1) → 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3082 ⊆ wss 3823 ‘cfv 6185 1c1 10334 Cℋ cch 28497 Statescst 28530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-iota 6149 df-fv 6193 |
This theorem is referenced by: stcltr2i 29845 stcltrlem2 29847 |
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