Theorem lrrecpo 27904

Description: Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.)

Hypothesis

Ref	Expression
lrrec.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}

Assertion

Ref	Expression
lrrecpo	⊢ 𝑅 Po No

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem lrrecpo

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayelon 27735	. . . . . 6 ⊢ ( bday ‘𝑎) ∈ On
2	1	onirri 6428	. . . . 5 ⊢ ¬ ( bday ‘𝑎) ∈ ( bday ‘𝑎)
3		lrrec.1	. . . . . . 7 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
4	3	lrrecval2 27903	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑎 ∈ No ) → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎)))
5	4	anidms 566	. . . . 5 ⊢ (𝑎 ∈ No → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎)))
6	2, 5	mtbiri 327	. . . 4 ⊢ (𝑎 ∈ No → ¬ 𝑎𝑅𝑎)
7	6	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑎 ∈ No ) → ¬ 𝑎𝑅𝑎)
8		bdayelon 27735	. . . . . 6 ⊢ ( bday ‘𝑐) ∈ On
9		ontr1 6361	. . . . . 6 ⊢ (( bday ‘𝑐) ∈ On → ((( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐)) → ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
10	8, 9	ax-mp 5	. . . . 5 ⊢ ((( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐)) → ( bday ‘𝑎) ∈ ( bday ‘𝑐))
11	3	lrrecval2 27903	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ) → (𝑎𝑅𝑏 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑏)))
12	11	3adant3 1132	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑏 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑏)))
13	3	lrrecval2 27903	. . . . . . . 8 ⊢ ((𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑏𝑅𝑐 ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))
14	13	3adant1 1130	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑏𝑅𝑐 ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))
15	12, 14	anbi12d 632	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) ↔ (( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐))))
16	3	lrrecval2 27903	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑐 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
17	16	3adant2 1131	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑐 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
18	15, 17	imbi12d 344	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) ↔ ((( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐)) → ( bday ‘𝑎) ∈ ( bday ‘𝑐))))
19	10, 18	mpbiri 258	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
20	19	adantl 481	. . 3 ⊢ ((⊤ ∧ (𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No )) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
21	7, 20	ispod 5538	. 2 ⊢ (⊤ → 𝑅 Po No )
22	21	mptru 1548	1 ⊢ 𝑅 Po No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113   ∪ cun 3896   class class class wbr 5095  {copab 5157   Po wpo 5527  Oncon0 6314  ‘cfv 6489   No csur 27598   bday cbday 27600   L cleft 27806   R cright 27807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-1o 8394  df-2o 8395  df-no 27601  df-slt 27602  df-bday 27603  df-sslt 27741  df-scut 27743  df-made 27808  df-old 27809  df-left 27811  df-right 27812

This theorem is referenced by:  noinds  27908  norecfn  27909  norecov  27910  noxpordpo  27913  no2indslem  27917  no3inds  27921