Theorem xpord2ind 8153

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 2728	. 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 8152	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  ∀wral 3058   class class class wbr 5148  {copab 5210   Po wpo 5588   Fr wfr 5630   Se wse 5631   × cxp 5676  Predcpred 6304  ‘cfv 6548  1^st c1st 7991  2^nd c2nd 7992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-fr 5633  df-se 5634  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-iota 6500  df-fun 6550  df-fv 6556  df-1st 7993  df-2nd 7994

This theorem is referenced by:  on2ind  8690  no2indslem  27884