Theorem xpord2ind 8150

Description: Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)

Hypotheses

Ref	Expression
xpord2ind.1	⊢ 𝑅 Fr 𝐴
xpord2ind.2	⊢ 𝑅 Po 𝐴
xpord2ind.3	⊢ 𝑅 Se 𝐴
xpord2ind.4	⊢ 𝑆 Fr 𝐵
xpord2ind.5	⊢ 𝑆 Po 𝐵
xpord2ind.6	⊢ 𝑆 Se 𝐵
xpord2ind.7	⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))
xpord2ind.8	⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))
xpord2ind.9	⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))
xpord2ind.11	⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))
xpord2ind.12	⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))
xpord2ind.i	⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))

Assertion

Ref	Expression
xpord2ind	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑   𝜓,𝑎   𝜏,𝑎   𝐵,𝑎,𝑏,𝑐,𝑑   𝜒,𝑏   𝜂,𝑏   𝜑,𝑐   𝜃,𝑐   𝜓,𝑑   𝑅,𝑎,𝑏,𝑐,𝑑   𝑆,𝑎,𝑏,𝑐,𝑑   𝑋,𝑎,𝑏   𝑌,𝑏

Allowed substitution hints:   𝜑(𝑎, 𝑏, 𝑑)   𝜓(𝑏, 𝑐)   𝜒(𝑎, 𝑐, 𝑑)   𝜃(𝑎, 𝑏, 𝑑)   𝜏(𝑏, 𝑐, 𝑑)   𝜂(𝑎, 𝑐, 𝑑)   𝑋(𝑐, 𝑑)   𝑌(𝑎, 𝑐, 𝑑)

Proof of Theorem xpord2ind

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
2		xpord2ind.1	. 2 ⊢ 𝑅 Fr 𝐴
3		xpord2ind.2	. 2 ⊢ 𝑅 Po 𝐴
4		xpord2ind.3	. 2 ⊢ 𝑅 Se 𝐴
5		xpord2ind.4	. 2 ⊢ 𝑆 Fr 𝐵
6		xpord2ind.5	. 2 ⊢ 𝑆 Po 𝐵
7		xpord2ind.6	. 2 ⊢ 𝑆 Se 𝐵
8		xpord2ind.7	. 2 ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))
9		xpord2ind.8	. 2 ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))
10		xpord2ind.9	. 2 ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))
11		xpord2ind.11	. 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))
12		xpord2ind.12	. 2 ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))
13		xpord2ind.i	. 2 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	xpord2indlem 8149	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   class class class wbr 5111  {copab 5175   Po wpo 5569   Fr wfr 5613   Se wse 5614   × cxp 5661  Predcpred 6305  ‘cfv 6540  1^st c1st 7990  2^nd c2nd 7991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-po 5571  df-fr 5616  df-se 5617  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fun 6542  df-fv 6548  df-1st 7992  df-2nd 7993

This theorem is used by:  on2ind  8661  no2indlesm  28200  ons2ind  28521