Theorem lrrecpo 33727

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 33604	. . . . . 6 ⊢ ( bday ‘𝑎) ∈ On
2	1	onirri 6273	. . . . 5 ⊢ ¬ ( bday ‘𝑎) ∈ ( bday ‘𝑎)
3		lrrec.1	. . . . . . 7 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
4	3	lrrecval2 33726	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑎 ∈ No ) → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎)))
5	4	anidms 570	. . . . 5 ⊢ (𝑎 ∈ No → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎)))
6	2, 5	mtbiri 330	. . . 4 ⊢ (𝑎 ∈ No → ¬ 𝑎𝑅𝑎)
7	6	adantl 485	. . 3 ⊢ ((⊤ ∧ 𝑎 ∈ No ) → ¬ 𝑎𝑅𝑎)
8		bdayelon 33604	. . . . . 6 ⊢ ( bday ‘𝑐) ∈ On
9		ontr1 6212	. . . . . 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 33726	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ) → (𝑎𝑅𝑏 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑏)))
12	11	3adant3 1133	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑏 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑏)))
13	3	lrrecval2 33726	. . . . . . . 8 ⊢ ((𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑏𝑅𝑐 ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))
14	13	3adant1 1131	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑏𝑅𝑐 ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))
15	12, 14	anbi12d 634	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) ↔ (( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐))))
16	3	lrrecval2 33726	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑐 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
17	16	3adant2 1132	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (𝑎𝑅𝑐 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑐)))
18	15, 17	imbi12d 348	. . . . 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 485	. . 3 ⊢ ((⊤ ∧ (𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No )) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
21	7, 20	ispod 5446	. 2 ⊢ (⊤ → 𝑅 Po No )
22	21	mptru 1549	1 ⊢ 𝑅 Po No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542  ⊤wtru 1543   ∈ wcel 2113   ∪ cun 3839   class class class wbr 5027  {copab 5089   Po wpo 5436  Oncon0 6166  ‘cfv 6333   No csur 33476   bday cbday 33478   L cleft 33662   R cright 33663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-wrecs 7969  df-recs 8030  df-1o 8124  df-2o 8125  df-no 33479  df-slt 33480  df-bday 33481  df-sslt 33609  df-scut 33611  df-made 33664  df-old 33665  df-left 33667  df-right 33668

This theorem is referenced by:  noinds  33731  norecfn  33732  norecov  33733  noxpordpo  33736  no2indslem  33740  no3indslem  33744