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Mirrors > Home > MPE Home > Th. List > xpord3ind | Structured version Visualization version GIF version |
Description: Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.) |
Ref | Expression |
---|---|
xpord3ind.1 | ⊢ 𝑅 Fr 𝐴 |
xpord3ind.2 | ⊢ 𝑅 Po 𝐴 |
xpord3ind.3 | ⊢ 𝑅 Se 𝐴 |
xpord3ind.4 | ⊢ 𝑆 Fr 𝐵 |
xpord3ind.5 | ⊢ 𝑆 Po 𝐵 |
xpord3ind.6 | ⊢ 𝑆 Se 𝐵 |
xpord3ind.7 | ⊢ 𝑇 Fr 𝐶 |
xpord3ind.8 | ⊢ 𝑇 Po 𝐶 |
xpord3ind.9 | ⊢ 𝑇 Se 𝐶 |
xpord3ind.10 | ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) |
xpord3ind.11 | ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) |
xpord3ind.12 | ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) |
xpord3ind.13 | ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) |
xpord3ind.14 | ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) |
xpord3ind.15 | ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) |
xpord3ind.16 | ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) |
xpord3ind.17 | ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) |
xpord3ind.18 | ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) |
xpord3ind.19 | ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) |
xpord3ind.i | ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) |
Ref | Expression |
---|---|
xpord3ind | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1136 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑋 ∈ 𝐴) | |
2 | simp2 1137 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑌 ∈ 𝐵) | |
3 | simp3 1138 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑍 ∈ 𝐶) | |
4 | xpord3ind.1 | . . 3 ⊢ 𝑅 Fr 𝐴 | |
5 | ax-1 6 | . . 3 ⊢ (𝑅 Fr 𝐴 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Fr 𝐴)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Fr 𝐴) |
7 | xpord3ind.2 | . . 3 ⊢ 𝑅 Po 𝐴 | |
8 | 7 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Po 𝐴) |
9 | xpord3ind.3 | . . 3 ⊢ 𝑅 Se 𝐴 | |
10 | 9 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑅 Se 𝐴) |
11 | xpord3ind.4 | . . 3 ⊢ 𝑆 Fr 𝐵 | |
12 | 11 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑆 Fr 𝐵) |
13 | xpord3ind.5 | . . 3 ⊢ 𝑆 Po 𝐵 | |
14 | 13 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑆 Po 𝐵) |
15 | xpord3ind.6 | . . 3 ⊢ 𝑆 Se 𝐵 | |
16 | 15 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑆 Se 𝐵) |
17 | xpord3ind.7 | . . 3 ⊢ 𝑇 Fr 𝐶 | |
18 | 17 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑇 Fr 𝐶) |
19 | xpord3ind.8 | . . 3 ⊢ 𝑇 Po 𝐶 | |
20 | 19 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑇 Po 𝐶) |
21 | xpord3ind.9 | . . 3 ⊢ 𝑇 Se 𝐶 | |
22 | 21 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝑇 Se 𝐶) |
23 | xpord3ind.10 | . 2 ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) | |
24 | xpord3ind.11 | . 2 ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) | |
25 | xpord3ind.12 | . 2 ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) | |
26 | xpord3ind.13 | . 2 ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) | |
27 | xpord3ind.14 | . 2 ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) | |
28 | xpord3ind.15 | . 2 ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) | |
29 | xpord3ind.16 | . 2 ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) | |
30 | xpord3ind.17 | . 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) | |
31 | xpord3ind.18 | . 2 ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) | |
32 | xpord3ind.19 | . 2 ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) | |
33 | xpord3ind.i | . . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) | |
34 | 33 | adantl 482 | . 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) |
35 | 1, 2, 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34 | xpord3indd 8087 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Po wpo 5543 Fr wfr 5585 Se wse 5586 Predcpred 6252 |
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-3or 1088 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-ot 4595 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: on3ind 8616 no3inds 27270 |
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