Theorem ltlecasei 11314

Description: Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
ltlecasei.1	⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝜓)
ltlecasei.2	⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓)
ltlecasei.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
ltlecasei.4	⊢ (𝜑 → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
ltlecasei	⊢ (𝜑 → 𝜓)

Proof of Theorem ltlecasei

Step	Hyp	Ref	Expression
1		ltlecasei.2	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓)
2		ltlecasei.1	. 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝜓)
3		ltlecasei.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		ltlecasei.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5		lelttric 11313	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ∨ 𝐴 < 𝐵))
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → (𝐵 ≤ 𝐴 ∨ 𝐴 < 𝐵))
7	1, 2, 6	mpjaodan 973	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∈ wcel 2149   class class class wbr 5110  ℝcr 11095   < clt 11239   ≤ cle 11240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-cnv 5667  df-xr 11243  df-le 11245

This theorem is referenced by:  iccsplit  13508  expnbnd  14264  hashf1  14490  absmax  15377  sinltx  16241  iccntr  24944  pmltpclem2  25573  cniccbdd  25585  iccvolcl  25691  ioovolcl  25694  dyaddisjlem  25719  mbfposr  25776  itg1ge0a  25835  itg2monolem1  25874  itgioo  25940  c1lip1  26121  plyeq0lem  26332  aalioulem5  26462  pserulm  26547  tanord  26665  birthdaylem3  27080  fsumharmonic  27138  chpo1ubb  27607  cos9thpiminplylem1  34113  mblfinlem2  38192  ioodvbdlimc1  46532  ioodvbdlimc2  46534  ibliooicc  46570  fourierdlem107  46812