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Mirrors > Home > ILE Home > Th. List > poleloe | GIF version |
Description: Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
poleloe | ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brun 4018 | . 2 ⊢ (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 I 𝐵)) | |
2 | ideqg 4740 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 2 | orbi2d 780 | . 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐴𝑅𝐵 ∨ 𝐴 I 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))) |
4 | 1, 3 | syl5bb 191 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 698 = wceq 1335 ∈ wcel 2128 ∪ cun 3100 class class class wbr 3967 I cid 4251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 |
This theorem is referenced by: poltletr 4989 |
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