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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 4027 | . 2 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) | |
2 | psstr 4032 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
3 | 2 | ex 416 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) |
4 | psseq2 4016 | . . . . 5 ⊢ (𝐵 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ 𝐶)) | |
5 | 4 | biimpcd 252 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊊ 𝐶)) |
6 | 3, 5 | jaod 856 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ((𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ⊊ 𝐶)) |
7 | 6 | imp 410 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) → 𝐴 ⊊ 𝐶) |
8 | 1, 7 | sylan2b 596 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ⊆ wss 3881 ⊊ wpss 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-in 3888 df-ss 3898 df-pss 3900 |
This theorem is referenced by: psssstrd 4037 suplem1pr 10463 atexch 30164 bj-2upln0 34459 bj-2upln1upl 34460 |
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