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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 2726 | . 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 8133 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 class class class wbr 5141 {copab 5203 Po wpo 5579 Fr wfr 5621 Se wse 5622 × cxp 5667 Predcpred 6293 ‘cfv 6537 1st c1st 7972 2nd c2nd 7973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-fr 5624 df-se 5625 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-iota 6489 df-fun 6539 df-fv 6545 df-1st 7974 df-2nd 7975 |
This theorem is referenced by: on2ind 8670 no2indslem 27826 |
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