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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 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 8079 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 class class class wbr 5105 {copab 5167 Po wpo 5543 Fr wfr 5585 Se wse 5586 × cxp 5631 Predcpred 6252 ‘cfv 6496 1st c1st 7919 2nd c2nd 7920 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-fr 5588 df-se 5589 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-iota 6448 df-fun 6498 df-fv 6504 df-1st 7921 df-2nd 7922 |
This theorem is referenced by: on2ind 8615 no2indslem 27266 |
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