Theorem xpord2ind 8098

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 2736	. 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 8097	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051   class class class wbr 5085  {copab 5147   Po wpo 5537   Fr wfr 5581   Se wse 5582   × cxp 5629  Predcpred 6264  ‘cfv 6498  1^st c1st 7940  2^nd c2nd 7941

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-fr 5584  df-se 5585  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-fun 6500  df-fv 6506  df-1st 7942  df-2nd 7943

This theorem is referenced by:  on2ind  8605  no2indlesm  27946  ons2ind  28267