Theorem lrrecpo 28096

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		bdayon 27907	. . . . . 6 ⊢ ( bday ‘𝑎) ∈ On
2	1	onirri 6473	. . . . 5 ⊢ ¬ ( bday ‘𝑎) ∈ ( bday ‘𝑎)
3		lrrec.1	. . . . . . 7 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
4	3	lrrecval2 28095	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑎 ∈ No ) → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎)))
5	4	anidms 576	. . . . 5 ⊢ (𝑎 ∈ No → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎)))
6	2, 5	mtbiri 330	. . . 4 ⊢ (𝑎 ∈ No → ¬ 𝑎𝑅𝑎)
7	6	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑎 ∈ No ) → ¬ 𝑎𝑅𝑎)
8		bdayon 27907	. . . . . 6 ⊢ ( bday ‘𝑐) ∈ On
9		ontr1 6406	. . . . . 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 28095	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ) → (𝑎𝑅𝑏 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑏)))
12	11	3adant3 1148	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑏 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑏)))
13	3	lrrecval2 28095	. . . . . . . 8 ⊢ ((𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑏𝑅𝑐 ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))
14	13	3adant1 1146	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑏𝑅𝑐 ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))
15	12, 14	anbi12d 643	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) ↔ (( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐))))
16	3	lrrecval2 28095	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑐 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
17	16	3adant2 1147	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑐 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
18	15, 17	imbi12d 347	. . . . 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 261	. . . 4 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
20	19	adantl 486	. . 3 ⊢ ((⊤ ∧ (𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No )) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
21	7, 20	ispod 5576	. 2 ⊢ (⊤ → 𝑅 Po No )
22	21	mptru 1574	1 ⊢ 𝑅 Po No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149   ∪ cun 3911   class class class wbr 5110  {copab 5174   Po wpo 5565  Oncon0 6358  ‘cfv 6534   No csur 27766   bday cbday 27768   L cleft 27980   R cright 27981

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-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-no 27769  df-lts 27770  df-bday 27771  df-slts 27913  df-cuts 27915  df-made 27982  df-old 27983  df-left 27985  df-right 27986

This theorem is referenced by:  noinds  28100  norecfn  28101  norecov  28102  noxpordpo  28105  no2indlesm  28109  no3inds  28113