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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 27528 | . . 3 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
2 | breq1 5150 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑦 ↔ 𝑋 <s 𝑦)) | |
3 | breq2 5151 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 <s 𝑦 ↔ 𝑋 <s 𝑌)) | |
4 | 2, 3 | rspc2va 3622 | . . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝑋 <s 𝑌) |
5 | 4 | ancoms 457 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 <s 𝑌) |
6 | 1, 5 | sylan 578 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 <s 𝑌) |
7 | 6 | 3impb 1113 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 ∈ wcel 2104 ∀wral 3059 class class class wbr 5147 <s cslt 27380 <<s csslt 27518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-sslt 27519 |
This theorem is referenced by: ssltsepcd 27532 ssltun1 27546 ssltun2 27547 ssltdisj 27559 cofcutr 27649 |
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