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Theorem xpord2ind 8148

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) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑆(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑆(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘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 8147	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 class class class wbr 5143 {copab 5205 Po wpo 5583 Fr wfr 5625 Se wse 5626 × cxp 5671 Predcpred 6299 ‘cfv 6543 1^st c1st 7986 2^nd c2nd 7987

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 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735

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 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-fr 5628 df-se 5629 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7988 df-2nd 7989

This theorem is referenced by: on2ind 8684 no2indslem 27865

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