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Mirrors > Home > MPE Home > Th. List > 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 27119 | . . . . . 6 ⊢ ( bday ‘𝑎) ∈ On | |
2 | 1 | onirri 6431 | . . . . 5 ⊢ ¬ ( bday ‘𝑎) ∈ ( bday ‘𝑎) |
3 | lrrec.1 | . . . . . . 7 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | |
4 | 3 | lrrecval2 27255 | . . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑎 ∈ No ) → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎))) |
5 | 4 | anidms 568 | . . . . 5 ⊢ (𝑎 ∈ No → (𝑎𝑅𝑎 ↔ ( bday ‘𝑎) ∈ ( bday ‘𝑎))) |
6 | 2, 5 | mtbiri 327 | . . . 4 ⊢ (𝑎 ∈ No → ¬ 𝑎𝑅𝑎) |
7 | 6 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑎 ∈ No ) → ¬ 𝑎𝑅𝑎) |
8 | bdayelon 27119 | . . . . . 6 ⊢ ( bday ‘𝑐) ∈ On | |
9 | ontr1 6364 | . . . . . 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 27255 | . . . . . . . 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 27255 | . . . . . . . 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 632 | . . . . . 6 ⊢ ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) ↔ (( bday ‘𝑎) ∈ ( bday ‘𝑏) ∧ ( bday ‘𝑏) ∈ ( bday ‘𝑐)))) |
16 | 3 | lrrecval2 27255 | . . . . . . 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 345 | . . . . 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 483 | . . 3 ⊢ ((⊤ ∧ (𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No )) → ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
21 | 7, 20 | ispod 5555 | . 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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ∪ cun 3909 class class class wbr 5106 {copab 5168 Po wpo 5544 Oncon0 6318 ‘cfv 6497 No csur 26991 bday cbday 26993 L cleft 27178 R cright 27179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-1o 8413 df-2o 8414 df-no 26994 df-slt 26995 df-bday 26996 df-sslt 27124 df-scut 27126 df-made 27180 df-old 27181 df-left 27183 df-right 27184 |
This theorem is referenced by: noinds 27260 norecfn 27261 norecov 27262 noxpordpo 27265 no2indslem 27269 no3inds 27273 |
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