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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrrecpo | Structured version Visualization version GIF version |
Description: Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
lrrec.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} |
Ref | Expression |
---|---|
lrrecpo | ⊢ 𝑅 Po No |
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 |
Colors of variables: wff setvar class |
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 |
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