Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4073 | . 2 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) | |
2 | psstr 4078 | . . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | |
3 | 2 | ex 413 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶)) |
4 | psseq2 4062 | . . . . 5 ⊢ (𝐵 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ 𝐶)) | |
5 | 4 | biimpcd 250 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 = 𝐶 → 𝐴 ⊊ 𝐶)) |
6 | 3, 5 | jaod 853 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ((𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶) → 𝐴 ⊊ 𝐶)) |
7 | 6 | imp 407 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶)) → 𝐴 ⊊ 𝐶) |
8 | 1, 7 | sylan2b 593 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ⊆ wss 3933 ⊊ wpss 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ne 3014 df-in 3940 df-ss 3949 df-pss 3951 |
This theorem is referenced by: psssstrd 4083 suplem1pr 10462 atexch 30085 bj-2upln0 34232 bj-2upln1upl 34233 |
Copyright terms: Public domain | W3C validator |