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Theorem xpord3indd 8179
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 2735 . 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 8178 1 (𝜅𝜆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059   class class class wbr 5148  {copab 5210   Po wpo 5595   Fr wfr 5638   Se wse 5639   × cxp 5687  Predcpred 6322  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  xpord3ind  8180
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