![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . 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 8119 | 1 ⊢ (𝜅 → 𝜆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 class class class wbr 5138 {copab 5200 Po wpo 5576 Fr wfr 5618 Se wse 5619 × cxp 5664 Predcpred 6285 ‘cfv 6529 1st c1st 7952 2nd c2nd 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-ot 4628 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-fr 5621 df-se 5622 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-iota 6481 df-fun 6531 df-fv 6537 df-1st 7954 df-2nd 7955 |
This theorem is referenced by: xpord3ind 8121 |
Copyright terms: Public domain | W3C validator |