Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssltsepcd | Structured version Visualization version GIF version |
Description: Two elements of separated sets obey less than. Deduction form of ssltsepc 33673. (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 33673 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → 𝑋 <s 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 class class class wbr 5039 <s cslt 33530 <<s csslt 33661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-sslt 33662 |
This theorem is referenced by: sslttr 33687 cofsslt 33774 coinitsslt 33775 cofcutrtime 33779 |
Copyright terms: Public domain | W3C validator |