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| Mirrors > Home > MPE Home > Th. List > xpord3indd | Structured version Visualization version GIF version | ||
| Description: Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| xpord3indd.x | ⊢ (𝜅 → 𝑋 ∈ 𝐴) | 
| xpord3indd.y | ⊢ (𝜅 → 𝑌 ∈ 𝐵) | 
| xpord3indd.z | ⊢ (𝜅 → 𝑍 ∈ 𝐶) | 
| xpord3indd.1 | ⊢ (𝜅 → 𝑅 Fr 𝐴) | 
| xpord3indd.2 | ⊢ (𝜅 → 𝑅 Po 𝐴) | 
| xpord3indd.3 | ⊢ (𝜅 → 𝑅 Se 𝐴) | 
| xpord3indd.4 | ⊢ (𝜅 → 𝑆 Fr 𝐵) | 
| xpord3indd.5 | ⊢ (𝜅 → 𝑆 Po 𝐵) | 
| xpord3indd.6 | ⊢ (𝜅 → 𝑆 Se 𝐵) | 
| xpord3indd.7 | ⊢ (𝜅 → 𝑇 Fr 𝐶) | 
| xpord3indd.8 | ⊢ (𝜅 → 𝑇 Po 𝐶) | 
| xpord3indd.9 | ⊢ (𝜅 → 𝑇 Se 𝐶) | 
| xpord3indd.10 | ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) | 
| xpord3indd.11 | ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) | 
| xpord3indd.12 | ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) | 
| xpord3indd.13 | ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) | 
| xpord3indd.14 | ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) | 
| xpord3indd.15 | ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) | 
| xpord3indd.16 | ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) | 
| xpord3indd.17 | ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) | 
| xpord3indd.18 | ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) | 
| xpord3indd.19 | ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) | 
| xpord3indd.i | ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) | 
| Ref | Expression | 
|---|---|
| xpord3indd | ⊢ (𝜅 → 𝜆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))} | |
| 2 | xpord3indd.x | . 2 ⊢ (𝜅 → 𝑋 ∈ 𝐴) | |
| 3 | xpord3indd.y | . 2 ⊢ (𝜅 → 𝑌 ∈ 𝐵) | |
| 4 | xpord3indd.z | . 2 ⊢ (𝜅 → 𝑍 ∈ 𝐶) | |
| 5 | xpord3indd.1 | . 2 ⊢ (𝜅 → 𝑅 Fr 𝐴) | |
| 6 | xpord3indd.2 | . 2 ⊢ (𝜅 → 𝑅 Po 𝐴) | |
| 7 | xpord3indd.3 | . 2 ⊢ (𝜅 → 𝑅 Se 𝐴) | |
| 8 | xpord3indd.4 | . 2 ⊢ (𝜅 → 𝑆 Fr 𝐵) | |
| 9 | xpord3indd.5 | . 2 ⊢ (𝜅 → 𝑆 Po 𝐵) | |
| 10 | xpord3indd.6 | . 2 ⊢ (𝜅 → 𝑆 Se 𝐵) | |
| 11 | xpord3indd.7 | . 2 ⊢ (𝜅 → 𝑇 Fr 𝐶) | |
| 12 | xpord3indd.8 | . 2 ⊢ (𝜅 → 𝑇 Po 𝐶) | |
| 13 | xpord3indd.9 | . 2 ⊢ (𝜅 → 𝑇 Se 𝐶) | |
| 14 | xpord3indd.10 | . 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) | |
| 15 | xpord3indd.11 | . 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) | |
| 16 | xpord3indd.12 | . 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) | |
| 17 | xpord3indd.13 | . 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) | |
| 18 | xpord3indd.14 | . 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) | |
| 19 | xpord3indd.15 | . 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) | |
| 20 | xpord3indd.16 | . 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) | |
| 21 | xpord3indd.17 | . 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) | |
| 22 | xpord3indd.18 | . 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) | |
| 23 | xpord3indd.19 | . 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) | |
| 24 | xpord3indd.i | . 2 ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) | |
| 25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 | xpord3inddlem 8180 | 1 ⊢ (𝜅 → 𝜆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 class class class wbr 5142 {copab 5204 Po wpo 5589 Fr wfr 5633 Se wse 5634 × cxp 5682 Predcpred 6319 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-ot 4634 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-fr 5636 df-se 5637 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fun 6562 df-fv 6568 df-1st 8015 df-2nd 8016 | 
| This theorem is referenced by: xpord3ind 8182 | 
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