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Mirrors > Home > MPE Home > Th. List > xpord2ind | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
xpord2ind | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂) |
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) |
Colors of variables: wff setvar class |
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 1st c1st 7991 2nd 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 |
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