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Theorem somin2 6141

Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

**Assertion**
Ref	Expression
somin2	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)

**Proof of Theorem somin2**
Step	Hyp	Ref	Expression
1		somincom 6140	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))
2		somin1 6139	. . 3 ⊢ ((𝑅 Or 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
3	2	ancom2s 648	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
4	1, 3	eqbrtrd 5170	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ∪ cun 3943 ifcif 4529 class class class wbr 5148 I cid 5574 Or wor 5588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3948 df-un 3950 df-ss 3962 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684

This theorem is referenced by: soltmin 6142

	
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