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Mirrors > Home > MPE Home > Th. List > psstr | Structured version Visualization version GIF version |
Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
psstr | ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssss 4056 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | pssss 4056 | . . 3 ⊢ (𝐵 ⊊ 𝐶 → 𝐵 ⊆ 𝐶) | |
3 | 1, 2 | sylan9ss 3958 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊆ 𝐶) |
4 | pssn2lp 4062 | . . . 4 ⊢ ¬ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) | |
5 | psseq1 4048 | . . . . 5 ⊢ (𝐴 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵)) | |
6 | 5 | anbi1d 631 | . . . 4 ⊢ (𝐴 = 𝐶 → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) ↔ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶))) |
7 | 4, 6 | mtbiri 327 | . . 3 ⊢ (𝐴 = 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)) |
8 | 7 | con2i 139 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 = 𝐶) |
9 | dfpss2 4046 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
10 | 3, 8, 9 | sylanbrc 584 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ⊆ wss 3911 ⊊ wpss 3912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3446 df-in 3918 df-ss 3928 df-pss 3930 |
This theorem is referenced by: sspsstr 4066 psssstr 4067 psstrd 4068 porpss 7665 inf3lem5 9573 ltsopr 10973 |
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