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Mirrors > Home > MPE Home > Th. List > ssltsepc | Structured version Visualization version GIF version |
Description: Two elements of separated sets obey less-than. (Contributed by Scott Fenton, 20-Aug-2024.) |
Ref | Expression |
---|---|
ssltsepc | ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssltsep 27152 | . . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
2 | breq1 5109 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑦 ↔ 𝑋 <s 𝑦)) | |
3 | breq2 5110 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 <s 𝑦 ↔ 𝑋 <s 𝑌)) | |
4 | 2, 3 | rspc2va 3590 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌) |
5 | 4 | ancoms 460 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 <s 𝑌) |
6 | 1, 5 | sylan 581 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 <s 𝑌) |
7 | 6 | 3impb 1116 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ∀wral 3061 class class class wbr 5106 <s cslt 27005 <<s csslt 27142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-sslt 27143 |
This theorem is referenced by: ssltsepcd 27155 ssltun1 27169 ssltun2 27170 ssltdisj 27182 cofcutr 27265 |
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