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Mirrors > Home > MPE Home > Th. List > Mathboxes > finxp00 | Structured version Visualization version GIF version |
Description: Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.) |
Ref | Expression |
---|---|
finxp00 | ⊢ (∅↑↑𝑁) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finxpeq2 36571 | . . . 4 ⊢ (𝑛 = ∅ → (∅↑↑𝑛) = (∅↑↑∅)) | |
2 | 1 | eqeq1d 2732 | . . 3 ⊢ (𝑛 = ∅ → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑∅) = ∅)) |
3 | finxpeq2 36571 | . . . 4 ⊢ (𝑛 = 𝑚 → (∅↑↑𝑛) = (∅↑↑𝑚)) | |
4 | 3 | eqeq1d 2732 | . . 3 ⊢ (𝑛 = 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑚) = ∅)) |
5 | finxpeq2 36571 | . . . 4 ⊢ (𝑛 = suc 𝑚 → (∅↑↑𝑛) = (∅↑↑suc 𝑚)) | |
6 | 5 | eqeq1d 2732 | . . 3 ⊢ (𝑛 = suc 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑suc 𝑚) = ∅)) |
7 | finxpeq2 36571 | . . . 4 ⊢ (𝑛 = 𝑁 → (∅↑↑𝑛) = (∅↑↑𝑁)) | |
8 | 7 | eqeq1d 2732 | . . 3 ⊢ (𝑛 = 𝑁 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑁) = ∅)) |
9 | finxp0 36575 | . . 3 ⊢ (∅↑↑∅) = ∅ | |
10 | suceq 6429 | . . . . . . . . 9 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅) | |
11 | df-1o 8468 | . . . . . . . . 9 ⊢ 1o = suc ∅ | |
12 | 10, 11 | eqtr4di 2788 | . . . . . . . 8 ⊢ (𝑚 = ∅ → suc 𝑚 = 1o) |
13 | finxpeq2 36571 | . . . . . . . 8 ⊢ (suc 𝑚 = 1o → (∅↑↑suc 𝑚) = (∅↑↑1o)) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = (∅↑↑1o)) |
15 | finxp1o 36576 | . . . . . . 7 ⊢ (∅↑↑1o) = ∅ | |
16 | 14, 15 | eqtrdi 2786 | . . . . . 6 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = ∅) |
17 | 16 | adantl 480 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 = ∅) → (∅↑↑suc 𝑚) = ∅) |
18 | finxpsuc 36582 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ((∅↑↑𝑚) × ∅)) | |
19 | xp0 6156 | . . . . . 6 ⊢ ((∅↑↑𝑚) × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2786 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ∅) |
21 | 17, 20 | pm2.61dane 3027 | . . . 4 ⊢ (𝑚 ∈ ω → (∅↑↑suc 𝑚) = ∅) |
22 | 21 | a1d 25 | . . 3 ⊢ (𝑚 ∈ ω → ((∅↑↑𝑚) = ∅ → (∅↑↑suc 𝑚) = ∅)) |
23 | 2, 4, 6, 8, 9, 22 | finds 7891 | . 2 ⊢ (𝑁 ∈ ω → (∅↑↑𝑁) = ∅) |
24 | finxpnom 36585 | . 2 ⊢ (¬ 𝑁 ∈ ω → (∅↑↑𝑁) = ∅) | |
25 | 23, 24 | pm2.61i 182 | 1 ⊢ (∅↑↑𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∅c0 4321 × cxp 5673 suc csuc 6365 ωcom 7857 1oc1o 8461 ↑↑cfinxp 36567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-en 8942 df-fin 8945 df-finxp 36568 |
This theorem is referenced by: (None) |
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