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Mirrors > Home > MPE Home > Th. List > sspsstr | Structured version Visualization version GIF version |
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
Ref | Expression |
---|---|
sspsstr | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 3668 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | psstr 3673 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
3 | 2 | ex 449 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) |
4 | psseq1 3656 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | |
5 | 4 | biimprd 237 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) |
6 | 3, 5 | jaoi 393 | . . 3 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) |
7 | 6 | imp 444 | . 2 ⊢ (((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
8 | 1, 7 | sylanb 488 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ⊆ wss 3540 ⊊ wpss 3541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ne 2782 df-in 3547 df-ss 3554 df-pss 3556 |
This theorem is referenced by: sspsstrd 3677 ordtr2 5685 php 8029 canthp1lem2 9354 suplem1pr 9753 fbfinnfr 21455 ppiltx 24703 |
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