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Theorem pleval2 17567

Description: "Less than or equal to" in terms of "less than". (sspss 4027 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)

**Hypotheses**
Ref	Expression
pleval2.b	⊢ 𝐵 = (Base‘𝐾)
pleval2.l	⊢ ≤ = (le‘𝐾)
pleval2.s	⊢ < = (lt‘𝐾)

**Assertion**
Ref	Expression
pleval2	⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))

**Proof of Theorem pleval2**
Step	Hyp	Ref	Expression
1		pleval2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		pleval2.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		pleval2.s	. . . 4 ⊢ < = (lt‘𝐾)
4	1, 2, 3	pleval2i 17566	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))
5	4	3adant1 1127	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))
6	2, 3	pltle 17563	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌))
7	1, 2	posref 17553	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
8	7	3adant3 1129	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ 𝑋)
9		breq2 5034	. . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
10	8, 9	syl5ibcom 248	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
11	6, 10	jaod 856	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 < 𝑌 ∨ 𝑋 = 𝑌) → 𝑋 ≤ 𝑌))
12	5, 11	impbid 215	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 Posetcpo 17542 ltcplt 17543

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-proset 17530 df-poset 17548 df-plt 17560

This theorem is referenced by: pltletr 17573 plelttr 17574 tosso 17638 tlt3 30678 orngsqr 30928

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