Theorem xpord2ind 8090

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 8089	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051   class class class wbr 5098  {copab 5160   Po wpo 5530   Fr wfr 5574   Se wse 5575   × cxp 5622  Predcpred 6258  ‘cfv 6492  1^st c1st 7931  2^nd c2nd 7932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-fr 5577  df-se 5578  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7933  df-2nd 7934

This theorem is referenced by:  on2ind  8597  no2indlesm  27950  ons2ind  28271