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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 4098 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
| 2 | pssss 4098 | . . 3 ⊢ (𝐵 ⊊ 𝐶 → 𝐵 ⊆ 𝐶) | |
| 3 | 1, 2 | sylan9ss 3997 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊆ 𝐶) |
| 4 | pssn2lp 4104 | . . . 4 ⊢ ¬ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) | |
| 5 | psseq1 4090 | . . . . 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 4088 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
| 10 | 3, 8, 9 | sylanbrc 583 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ⊆ wss 3951 ⊊ wpss 3952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 df-ss 3968 df-pss 3971 |
| This theorem is referenced by: sspsstr 4108 psssstr 4109 psstrd 4110 porpss 7747 inf3lem5 9672 ltsopr 11072 |
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