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Theorem xpord3ind 8146
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, 4-Sep-2024.)
Hypotheses
Ref Expression
xpord3ind.1 𝑅 Fr 𝐴
xpord3ind.2 𝑅 Po 𝐴
xpord3ind.3 𝑅 Se 𝐴
xpord3ind.4 𝑆 Fr 𝐵
xpord3ind.5 𝑆 Po 𝐵
xpord3ind.6 𝑆 Se 𝐵
xpord3ind.7 𝑇 Fr 𝐶
xpord3ind.8 𝑇 Po 𝐶
xpord3ind.9 𝑇 Se 𝐶
xpord3ind.10 (𝑎 = 𝑑 → (𝜑𝜓))
xpord3ind.11 (𝑏 = 𝑒 → (𝜓𝜒))
xpord3ind.12 (𝑐 = 𝑓 → (𝜒𝜃))
xpord3ind.13 (𝑎 = 𝑑 → (𝜏𝜃))
xpord3ind.14 (𝑏 = 𝑒 → (𝜂𝜏))
xpord3ind.15 (𝑏 = 𝑒 → (𝜁𝜃))
xpord3ind.16 (𝑐 = 𝑓 → (𝜎𝜏))
xpord3ind.17 (𝑎 = 𝑋 → (𝜑𝜌))
xpord3ind.18 (𝑏 = 𝑌 → (𝜌𝜇))
xpord3ind.19 (𝑐 = 𝑍 → (𝜇𝜆))
xpord3ind.i ((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
xpord3ind ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝜆)
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,𝑑,𝑒,𝑓   𝑌,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑍,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝜓,𝑎   𝜌,𝑎   𝜃,𝑎   𝜒,𝑏   𝜇,𝑏   𝜃,𝑏   𝜆,𝑐   𝜃,𝑐   𝜒,𝑓   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)

Proof of Theorem xpord3ind
StepHypRef Expression
1 simp1 1135 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑋𝐴)
2 simp2 1136 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑌𝐵)
3 simp3 1137 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑍𝐶)
4 xpord3ind.1 . . 3 𝑅 Fr 𝐴
5 ax-1 6 . . 3 (𝑅 Fr 𝐴 → ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑅 Fr 𝐴))
64, 5ax-mp 5 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑅 Fr 𝐴)
7 xpord3ind.2 . . 3 𝑅 Po 𝐴
87a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑅 Po 𝐴)
9 xpord3ind.3 . . 3 𝑅 Se 𝐴
109a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑅 Se 𝐴)
11 xpord3ind.4 . . 3 𝑆 Fr 𝐵
1211a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑆 Fr 𝐵)
13 xpord3ind.5 . . 3 𝑆 Po 𝐵
1413a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑆 Po 𝐵)
15 xpord3ind.6 . . 3 𝑆 Se 𝐵
1615a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑆 Se 𝐵)
17 xpord3ind.7 . . 3 𝑇 Fr 𝐶
1817a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑇 Fr 𝐶)
19 xpord3ind.8 . . 3 𝑇 Po 𝐶
2019a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑇 Po 𝐶)
21 xpord3ind.9 . . 3 𝑇 Se 𝐶
2221a1i 11 . 2 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝑇 Se 𝐶)
23 xpord3ind.10 . 2 (𝑎 = 𝑑 → (𝜑𝜓))
24 xpord3ind.11 . 2 (𝑏 = 𝑒 → (𝜓𝜒))
25 xpord3ind.12 . 2 (𝑐 = 𝑓 → (𝜒𝜃))
26 xpord3ind.13 . 2 (𝑎 = 𝑑 → (𝜏𝜃))
27 xpord3ind.14 . 2 (𝑏 = 𝑒 → (𝜂𝜏))
28 xpord3ind.15 . 2 (𝑏 = 𝑒 → (𝜁𝜃))
29 xpord3ind.16 . 2 (𝑐 = 𝑓 → (𝜎𝜏))
30 xpord3ind.17 . 2 (𝑎 = 𝑋 → (𝜑𝜌))
31 xpord3ind.18 . 2 (𝑏 = 𝑌 → (𝜌𝜇))
32 xpord3ind.19 . 2 (𝑐 = 𝑍 → (𝜇𝜆))
33 xpord3ind.i . . 3 ((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))
3433adantl 481 . 2 (((𝑋𝐴𝑌𝐵𝑍𝐶) ∧ (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))
351, 2, 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34xpord3indd 8145 1 ((𝑋𝐴𝑌𝐵𝑍𝐶) → 𝜆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086   = wceq 1540  wcel 2105  wral 3060   Po wpo 5586   Fr wfr 5628   Se wse 5629  Predcpred 6299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-fr 5631  df-se 5632  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7979  df-2nd 7980
This theorem is referenced by:  on3ind  8673  no3inds  27681
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