![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssltsepcd | Structured version Visualization version GIF version |
Description: Two elements of separated sets obey less-than. Deduction form of ssltsepc 27220. (Contributed by Scott Fenton, 25-Sep-2024.) |
Ref | Expression |
---|---|
ssltsepcd.1 | ⊢ (𝜑 → 𝐴 <<s 𝐵) |
ssltsepcd.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
ssltsepcd.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ssltsepcd | ⊢ (𝜑 → 𝑋 <s 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssltsepcd.1 | . 2 ⊢ (𝜑 → 𝐴 <<s 𝐵) | |
2 | ssltsepcd.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
3 | ssltsepcd.3 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ssltsepc 27220 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | 1 ⊢ (𝜑 → 𝑋 <s 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5141 <s cslt 27071 <<s csslt 27208 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-sslt 27209 |
This theorem is referenced by: sslttr 27234 cofsslt 27325 coinitsslt 27326 cofcutrtime 27334 addsproplem2 27370 addsproplem4 27372 addsproplem5 27373 addsproplem6 27374 addsuniflem 27400 negsproplem2 27419 negsproplem4 27421 negsproplem5 27422 negsproplem6 27423 negsunif 27443 mulsproplem5 27489 mulsproplem6 27490 mulsproplem7 27491 mulsproplem8 27492 mulsproplem12 27496 ssltmul1 27514 ssltmul2 27515 mulsuniflem 27516 |
Copyright terms: Public domain | W3C validator |