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Theorem xpord3indd 8181
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.)
Hypotheses
Ref Expression
xpord3indd.x (𝜅𝑋𝐴)
xpord3indd.y (𝜅𝑌𝐵)
xpord3indd.z (𝜅𝑍𝐶)
xpord3indd.1 (𝜅𝑅 Fr 𝐴)
xpord3indd.2 (𝜅𝑅 Po 𝐴)
xpord3indd.3 (𝜅𝑅 Se 𝐴)
xpord3indd.4 (𝜅𝑆 Fr 𝐵)
xpord3indd.5 (𝜅𝑆 Po 𝐵)
xpord3indd.6 (𝜅𝑆 Se 𝐵)
xpord3indd.7 (𝜅𝑇 Fr 𝐶)
xpord3indd.8 (𝜅𝑇 Po 𝐶)
xpord3indd.9 (𝜅𝑇 Se 𝐶)
xpord3indd.10 (𝑎 = 𝑑 → (𝜑𝜓))
xpord3indd.11 (𝑏 = 𝑒 → (𝜓𝜒))
xpord3indd.12 (𝑐 = 𝑓 → (𝜒𝜃))
xpord3indd.13 (𝑎 = 𝑑 → (𝜏𝜃))
xpord3indd.14 (𝑏 = 𝑒 → (𝜂𝜏))
xpord3indd.15 (𝑏 = 𝑒 → (𝜁𝜃))
xpord3indd.16 (𝑐 = 𝑓 → (𝜎𝜏))
xpord3indd.17 (𝑎 = 𝑋 → (𝜑𝜌))
xpord3indd.18 (𝑏 = 𝑌 → (𝜌𝜇))
xpord3indd.19 (𝑐 = 𝑍 → (𝜇𝜆))
xpord3indd.i ((𝜅 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (((∀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))
Assertion
Ref Expression
xpord3indd (𝜅𝜆)
Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑅,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑆,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑇,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑋,𝑎,𝑏,𝑐   𝑌,𝑏,𝑐   𝑍,𝑐   𝜅,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝜓,𝑎   𝜌,𝑎   𝜃,𝑎   𝜒,𝑏,𝑓   𝜇,𝑏   𝜃,𝑏   𝜆,𝑐   𝜃,𝑐   𝜑,𝑑   𝜏,𝑑   𝜂,𝑒   𝜓,𝑒   𝜁,𝑒   𝜎,𝑓
Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐)   𝜓(𝑓,𝑏,𝑐,𝑑)   𝜒(𝑒,𝑎,𝑐,𝑑)   𝜃(𝑒,𝑓,𝑑)   𝜏(𝑒,𝑓,𝑎,𝑏,𝑐)   𝜂(𝑓,𝑎,𝑏,𝑐,𝑑)   𝜁(𝑓,𝑎,𝑏,𝑐,𝑑)   𝜎(𝑒,𝑎,𝑏,𝑐,𝑑)   𝜌(𝑒,𝑓,𝑏,𝑐,𝑑)   𝜇(𝑒,𝑓,𝑎,𝑐,𝑑)   𝜆(𝑒,𝑓,𝑎,𝑏,𝑑)   𝑋(𝑒,𝑓,𝑑)   𝑌(𝑒,𝑓,𝑎,𝑑)   𝑍(𝑒,𝑓,𝑎,𝑏,𝑑)

Proof of Theorem xpord3indd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))}
2 xpord3indd.x . 2 (𝜅𝑋𝐴)
3 xpord3indd.y . 2 (𝜅𝑌𝐵)
4 xpord3indd.z . 2 (𝜅𝑍𝐶)
5 xpord3indd.1 . 2 (𝜅𝑅 Fr 𝐴)
6 xpord3indd.2 . 2 (𝜅𝑅 Po 𝐴)
7 xpord3indd.3 . 2 (𝜅𝑅 Se 𝐴)
8 xpord3indd.4 . 2 (𝜅𝑆 Fr 𝐵)
9 xpord3indd.5 . 2 (𝜅𝑆 Po 𝐵)
10 xpord3indd.6 . 2 (𝜅𝑆 Se 𝐵)
11 xpord3indd.7 . 2 (𝜅𝑇 Fr 𝐶)
12 xpord3indd.8 . 2 (𝜅𝑇 Po 𝐶)
13 xpord3indd.9 . 2 (𝜅𝑇 Se 𝐶)
14 xpord3indd.10 . 2 (𝑎 = 𝑑 → (𝜑𝜓))
15 xpord3indd.11 . 2 (𝑏 = 𝑒 → (𝜓𝜒))
16 xpord3indd.12 . 2 (𝑐 = 𝑓 → (𝜒𝜃))
17 xpord3indd.13 . 2 (𝑎 = 𝑑 → (𝜏𝜃))
18 xpord3indd.14 . 2 (𝑏 = 𝑒 → (𝜂𝜏))
19 xpord3indd.15 . 2 (𝑏 = 𝑒 → (𝜁𝜃))
20 xpord3indd.16 . 2 (𝑐 = 𝑓 → (𝜎𝜏))
21 xpord3indd.17 . 2 (𝑎 = 𝑋 → (𝜑𝜌))
22 xpord3indd.18 . 2 (𝑏 = 𝑌 → (𝜌𝜇))
23 xpord3indd.19 . 2 (𝑐 = 𝑍 → (𝜇𝜆))
24 xpord3indd.i . 2 ((𝜅 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (((∀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))
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24xpord3inddlem 8180 1 (𝜅𝜆)
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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