Theorem xpord3indd 8172

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	⊢ (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))

Assertion

Ref	Expression
xpord3indd	⊢ (𝜅 → 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   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.

Step	Hyp	Ref	Expression
1		eqid 2760	. 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 8171	1 ⊢ (𝜅 → 𝜆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   class class class wbr 5102  {copab 5166   Po wpo 5556   Fr wfr 5600   Se wse 5601   × cxp 5648  Predcpred 6303  ‘cfv 6538  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-ot 4592  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-fr 5603  df-se 5604  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002

This theorem is used by:  xpord3ind  8173