Theorem poleloe 6088

Description: Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
poleloe	⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)))

Proof of Theorem poleloe

Step	Hyp	Ref	Expression
1		brun 5137	. 2 ⊢ (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 I 𝐵))
2		ideqg 5800	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
3	2	orbi2d 916	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐴𝑅𝐵 ∨ 𝐴 I 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)))
4	1, 3	bitrid 283	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3888   class class class wbr 5086   I cid 5518

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631

This theorem is referenced by:  poltletr  6089  somin1  6090