Theorem lrrecpo 28384

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 28195	. . . . . 6 ⊢ (bday‘𝑎) ∈ On
2	1	onirri 6503	. . . . 5 ⊢ ¬ (bday‘𝑎) ∈ (bday‘𝑎)
3		lrrec.1	. . . . . . 7 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}
4	3	lrrecval2 28383	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑎 ∈ No) → (𝑎𝑅𝑎 ↔ (bday‘𝑎) ∈ (bday‘𝑎)))
5	4	anidms 577	. . . . 5 ⊢ (𝑎 ∈ No → (𝑎𝑅𝑎 ↔ (bday‘𝑎) ∈ (bday‘𝑎)))
6	2, 5	mtbiri 330	. . . 4 ⊢ (𝑎 ∈ No → ¬ 𝑎𝑅𝑎)
7	6	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑎 ∈ No) → ¬ 𝑎𝑅𝑎)
8		bdayon 28195	. . . . . 6 ⊢ (bday‘𝑐) ∈ On
9		ontr1 6436	. . . . . 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 28383	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No) → (𝑎𝑅𝑏 ↔ (bday‘𝑎) ∈ (bday‘𝑏)))
12	11	3adant3 1150	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No) → (𝑎𝑅𝑏 ↔ (bday‘𝑎) ∈ (bday‘𝑏)))
13	3	lrrecval2 28383	. . . . . . . 8 ⊢ ((𝑏 ∈ No ∧ 𝑐 ∈ No) → (𝑏𝑅𝑐 ↔ (bday‘𝑏) ∈ (bday‘𝑐)))
14	13	3adant1 1148	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No) → (𝑏𝑅𝑐 ↔ (bday‘𝑏) ∈ (bday‘𝑐)))
15	12, 14	anbi12d 644	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) ↔ ((bday‘𝑎) ∈ (bday‘𝑏) ∧ (bday‘𝑏) ∈ (bday‘𝑐))))
16	3	lrrecval2 28383	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ 𝑐 ∈ No) → (𝑎𝑅𝑐 ↔ (bday‘𝑎) ∈ (bday‘𝑐)))
17	16	3adant2 1149	. . . . . 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 487	. . 3 ⊢ ((⊤ ∧ (𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No)) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
21	7, 20	ispod 5575	. 2 ⊢ (⊤ → 𝑅 Po No)
22	21	mptru 1577	1 ⊢ 𝑅 Po No

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147   ∪ cun 4152   class class class wbr 5108  {copab 5173   Po wpo 5564  Oncon0 6388  ‘cfv 6566  Nocsur 28054  bdaycbday 28056  Lcleft 28268  Rcright 28269

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-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271  df-left 28273  df-right 28274

This theorem is used by:  noinds  28388  norecfn  28389  norecov  28390  noxpordpo  28393  no2indlesm  28397  no3inds  28401