Theorem xpord3ind 8110

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 → (𝜑 ↔ 𝜓))
xpord3ind.11	⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
xpord3ind.12	⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
xpord3ind.13	⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
xpord3ind.14	⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
xpord3ind.15	⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
xpord3ind.16	⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
xpord3ind.17	⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
xpord3ind.18	⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
xpord3ind.19	⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
xpord3ind.i	⊢ ((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
xpord3ind	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 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

Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑋 ∈ 𝐴)
2		simp2 1138	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑌 ∈ 𝐵)
3		simp3 1139	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑍 ∈ 𝐶)
4		xpord3ind.1	. . 3 ⊢ 𝑅 Fr 𝐴
5		ax-1 6	. . 3 ⊢ (𝑅 Fr 𝐴 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Fr 𝐴))
6	4, 5	ax-mp 5	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Fr 𝐴)
7		xpord3ind.2	. . 3 ⊢ 𝑅 Po 𝐴
8	7	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Po 𝐴)
9		xpord3ind.3	. . 3 ⊢ 𝑅 Se 𝐴
10	9	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Se 𝐴)
11		xpord3ind.4	. . 3 ⊢ 𝑆 Fr 𝐵
12	11	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑆 Fr 𝐵)
13		xpord3ind.5	. . 3 ⊢ 𝑆 Po 𝐵
14	13	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑆 Po 𝐵)
15		xpord3ind.6	. . 3 ⊢ 𝑆 Se 𝐵
16	15	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑆 Se 𝐵)
17		xpord3ind.7	. . 3 ⊢ 𝑇 Fr 𝐶
18	17	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑇 Fr 𝐶)
19		xpord3ind.8	. . 3 ⊢ 𝑇 Po 𝐶
20	19	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑇 Po 𝐶)
21		xpord3ind.9	. . 3 ⊢ 𝑇 Se 𝐶
22	21	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑇 Se 𝐶)
23		xpord3ind.10	. 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))
24		xpord3ind.11	. 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))
25		xpord3ind.12	. 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))
26		xpord3ind.13	. 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))
27		xpord3ind.14	. 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))
28		xpord3ind.15	. 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))
29		xpord3ind.16	. 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))
30		xpord3ind.17	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))
31		xpord3ind.18	. 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))
32		xpord3ind.19	. 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))
33		xpord3ind.i	. . 3 ⊢ ((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))
34	33	adantl 481	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) ∧ (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))
35	1, 2, 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34	xpord3indd 8109	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   Po wpo 5540   Fr wfr 5584   Se wse 5585  Predcpred 6268

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-ot 4591  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-po 5542  df-fr 5587  df-se 5588  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-iota 6458  df-fun 6504  df-fv 6510  df-1st 7945  df-2nd 7946

This theorem is referenced by:  on3ind  8610  no3inds  27971