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Mirrors > Home > MPE Home > Th. List > psssstr | Structured version Visualization version GIF version |
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
Ref | Expression |
---|---|
psssstr | ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 4060 | . 2 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) | |
2 | psstr 4065 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
3 | 2 | ex 414 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) |
4 | psseq2 4049 | . . . . 5 ⊢ (𝐵 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ 𝐶)) | |
5 | 4 | biimpcd 249 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊊ 𝐶)) |
6 | 3, 5 | jaod 858 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ((𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ⊊ 𝐶)) |
7 | 6 | imp 408 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) → 𝐴 ⊊ 𝐶) |
8 | 1, 7 | sylan2b 595 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = 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-or 847 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: psssstrd 4070 suplem1pr 10993 atexch 31365 bj-2upln0 35540 bj-2upln1upl 35541 |
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