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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 8160 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 class class class wbr 5152 {copab 5214 Po wpo 5592 Fr wfr 5634 Se wse 5635 × cxp 5680 Predcpred 6309 ‘cfv 6553 1st c1st 7999 2nd c2nd 8000 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-fr 5637 df-se 5638 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-iota 6505 df-fun 6555 df-fv 6561 df-1st 8001 df-2nd 8002 |
This theorem is referenced by: on2ind 8698 no2indslem 27899 |
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