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Theorem pltval 18233

Description: Less-than relation. (df-pss 3922 analog.) (Contributed by NM, 12-Oct-2011.)

**Hypotheses**
Ref	Expression
pltval.l	⊢ ≤ = (le‘𝐾)
pltval.s	⊢ < = (lt‘𝐾)

**Assertion**
Ref	Expression
pltval	⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))

**Proof of Theorem pltval**
Step	Hyp	Ref	Expression
1		pltval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
2		pltval.s	. . . . 5 ⊢ < = (lt‘𝐾)
3	1, 2	pltfval 18232	. . . 4 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))
4	3	breqd 5102	. . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋 < 𝑌 ↔ 𝑋( ≤ ∖ I )𝑌))
5		brdif 5144	. . . 4 ⊢ (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌))
6		ideqg 5791	. . . . . . 7 ⊢ (𝑌 ∈ 𝐶 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
7	6	necon3bbid 2965	. . . . . 6 ⊢ (𝑌 ∈ 𝐶 → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌))
8	7	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌))
9	8	anbi2d 630	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
10	5, 9	bitrid 283	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
11	4, 10	sylan9bb 509	. 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
12	11	3impb 1114	1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3899 class class class wbr 5091 I cid 5510 ‘cfv 6481 lecple 17165 ltcplt 18211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-iota 6437 df-fun 6483 df-fv 6489 df-plt 18231

This theorem is referenced by: pltle 18234 pltne 18235 pleval2i 18237 pltnle 18239 pltval3 18240 plttr 18243 latnlemlt 18375 latnle 18376 ipolt 18438 ogrpaddlt 20048 ogrpsublt 20052 ornglmullt 20782 orngrmullt 20783 orngmullt 20784 ofldlt1 20788 opltn0 39228 cvrval2 39312 cvrnbtwn2 39313 cvrnbtwn3 39314 cvrle 39316 cvrnbtwn4 39317 cvrne 39319 atlltn0 39344 hlrelat5N 39439 llnle 39556 lplnle 39578 llncvrlpln2 39595 lplncvrlvol2 39653 lhp2lt 40039 lautlt 40129

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