Theorem pltval3 18409

Description: Alternate expression for the "less than" relation. (dfpss3 4112 analog.) (Contributed by NM, 4-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
pltval3	⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))

Proof of Theorem pltval3

Step	Hyp	Ref	Expression
1		pleval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		pleval2.s	. . 3 ⊢ < = (lt‘𝐾)
3	1, 2	pltval 18402	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
4		pleval2.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	4, 1	posref 18388	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
6	5	3adant3 1132	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ 𝑋)
7		breq1 5169	. . . . . . 7 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
8	6, 7	syl5ibcom 245	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑌 ≤ 𝑋))
9	8	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 = 𝑌 → 𝑌 ≤ 𝑋))
10	4, 1	posasymb 18389	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
11	10	biimpd 229	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))
12	11	expdimp 452	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑌 ≤ 𝑋 → 𝑋 = 𝑌))
13	9, 12	impbid 212	. . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 = 𝑌 ↔ 𝑌 ≤ 𝑋))
14	13	necon3abid 2983	. . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ≠ 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))
15	14	pm5.32da 578	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
16	3, 15	bitrd 279	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   class class class wbr 5166  ‘cfv 6573  Basecbs 17258  lecple 17318  Posetcpo 18377  ltcplt 18378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-proset 18365  df-poset 18383  df-plt 18400

This theorem is referenced by:  tltnle  18492  opltcon3b  39160