Theorem xporderlem 6175

Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Hypothesis

Ref	Expression
xporderlem.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}

Assertion

Ref	Expression
xporderlem	⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑎,𝑦   𝑥,𝑏,𝑦   𝑥,𝑐,𝑦   𝑥,𝑑,𝑦

Allowed substitution hints:   𝐴(𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑐,𝑑)   𝑅(𝑎,𝑏,𝑐,𝑑)   𝑆(𝑎,𝑏,𝑐,𝑑)   𝑇(𝑥,𝑦,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem xporderlem

Step	Hyp	Ref	Expression
1		df-br 3966	. . 3 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ 𝑇)
2		xporderlem.1	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}
3	2	eleq2i 2224	. . 3 ⊢ (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ 𝑇 ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))})
4	1, 3	bitri 183	. 2 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))})
5		vex 2715	. . . 4 ⊢ 𝑎 ∈ V
6		vex 2715	. . . 4 ⊢ 𝑏 ∈ V
7	5, 6	opex 4189	. . 3 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
8		vex 2715	. . . 4 ⊢ 𝑐 ∈ V
9		vex 2715	. . . 4 ⊢ 𝑑 ∈ V
10	8, 9	opex 4189	. . 3 ⊢ ⟨𝑐, 𝑑⟩ ∈ V
11		eleq1 2220	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵)))
12		opelxp 4615	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))
13	11, 12	bitrdi 195	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
14	13	anbi1d 461	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵))))
15	5, 6	op1std 6093	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑥) = 𝑎)
16	15	breq1d 3975	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥)𝑅(1st ‘𝑦) ↔ 𝑎𝑅(1st ‘𝑦)))
17	15	eqeq1d 2166	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥) = (1st ‘𝑦) ↔ 𝑎 = (1st ‘𝑦)))
18	5, 6	op2ndd 6094	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑥) = 𝑏)
19	18	breq1d 3975	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ↔ 𝑏𝑆(2nd ‘𝑦)))
20	17, 19	anbi12d 465	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦)) ↔ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦))))
21	16, 20	orbi12d 783	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))) ↔ (𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦)))))
22	14, 21	anbi12d 465	. . 3 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦)))) ↔ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦))))))
23		eleq1 2220	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)))
24		opelxp 4615	. . . . . 6 ⊢ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵))
25	23, 24	bitrdi 195	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)))
26	25	anbi2d 460	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵))))
27	8, 9	op1std 6093	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑦) = 𝑐)
28	27	breq2d 3977	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1st ‘𝑦) ↔ 𝑎𝑅𝑐))
29	27	eqeq2d 2169	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎 = (1st ‘𝑦) ↔ 𝑎 = 𝑐))
30	8, 9	op2ndd 6094	. . . . . . 7 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
31	30	breq2d 3977	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2nd ‘𝑦) ↔ 𝑏𝑆𝑑))
32	29, 31	anbi12d 465	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦)) ↔ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))
33	28, 32	orbi12d 783	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦))) ↔ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
34	26, 33	anbi12d 465	. . 3 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦)))) ↔ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))))
35	7, 10, 22, 34	opelopab 4231	. 2 ⊢ (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ↔ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
36		an4 576	. . 3 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)))
37	36	anbi1i 454	. 2 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))) ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
38	4, 35, 37	3bitri 205	1 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1335   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965  {copab 4024   × cxp 4583  ‘cfv 5169  1^st c1st 6083  2^nd c2nd 6084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fv 5177  df-1st 6085  df-2nd 6086

This theorem is referenced by:  poxp  6176