Theorem tsrlemax 18789

Description: Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypothesis

Ref	Expression
istsr.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
tsrlemax	⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))

Proof of Theorem tsrlemax

Step	Hyp	Ref	Expression
1		breq2 5110	. . 3 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐶 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
2	1	bibi1d 346	. 2 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))))
3		breq2 5110	. . 3 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐵 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
4	3	bibi1d 346	. 2 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))))
5		olc 882	. . 3 ⊢ (𝐴𝑅𝐶 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))
6		eqid 2763	. . . . . . . . . 10 ⊢ dom 𝑅 = dom 𝑅
7	6	istsr 18786	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
8	7	simplbi 502	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
9		pstr 18780	. . . . . . . . 9 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
10	9	3expib 1140	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
11	8, 10	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
12	11	adantr 486	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
13	12	expdimp 458	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
14	13	impancom 457	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
15		idd 25	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐶))
16	14, 15	jaod 873	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐶))
17	5, 16	impbid2 229	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
18		orc 881	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))
19		idd 25	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐵))
20		istsr.1	. . . . . . . 8 ⊢ 𝑋 = dom 𝑅
21	20	tsrlin 18788	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
22	21	3adant3r1 1201	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
23	22	orcanai 1018	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
24		pstr 18780	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)
25	24	3expib 1140	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
26	8, 25	syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
27	26	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
28	27	expdimp 458	. . . . . 6 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐶) → (𝐶𝑅𝐵 → 𝐴𝑅𝐵))
29	28	impancom 457	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶𝑅𝐵) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵))
30	23, 29	syldan 603	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵))
31	19, 30	jaod 873	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐵))
32	18, 31	impbid2 229	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
33	2, 4, 17, 32	ifbothda 4524	1 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152  ifcif 4485   class class class wbr 5106   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  PosetRelcps 18767   TosetRel ctsr 18768

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-ps 18769  df-tsr 18770

This theorem is used by:  ordtbaslem  23536