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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 36173 | . . . 4 ⊢ (𝑛 = ∅ → (∅↑↑𝑛) = (∅↑↑∅)) | |
2 | 1 | eqeq1d 2735 | . . 3 ⊢ (𝑛 = ∅ → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑∅) = ∅)) |
3 | finxpeq2 36173 | . . . 4 ⊢ (𝑛 = 𝑚 → (∅↑↑𝑛) = (∅↑↑𝑚)) | |
4 | 3 | eqeq1d 2735 | . . 3 ⊢ (𝑛 = 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑚) = ∅)) |
5 | finxpeq2 36173 | . . . 4 ⊢ (𝑛 = suc 𝑚 → (∅↑↑𝑛) = (∅↑↑suc 𝑚)) | |
6 | 5 | eqeq1d 2735 | . . 3 ⊢ (𝑛 = suc 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑suc 𝑚) = ∅)) |
7 | finxpeq2 36173 | . . . 4 ⊢ (𝑛 = 𝑁 → (∅↑↑𝑛) = (∅↑↑𝑁)) | |
8 | 7 | eqeq1d 2735 | . . 3 ⊢ (𝑛 = 𝑁 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑁) = ∅)) |
9 | finxp0 36177 | . . 3 ⊢ (∅↑↑∅) = ∅ | |
10 | suceq 6422 | . . . . . . . . 9 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅) | |
11 | df-1o 8453 | . . . . . . . . 9 ⊢ 1o = suc ∅ | |
12 | 10, 11 | eqtr4di 2791 | . . . . . . . 8 ⊢ (𝑚 = ∅ → suc 𝑚 = 1o) |
13 | finxpeq2 36173 | . . . . . . . 8 ⊢ (suc 𝑚 = 1o → (∅↑↑suc 𝑚) = (∅↑↑1o)) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = (∅↑↑1o)) |
15 | finxp1o 36178 | . . . . . . 7 ⊢ (∅↑↑1o) = ∅ | |
16 | 14, 15 | eqtrdi 2789 | . . . . . 6 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = ∅) |
17 | 16 | adantl 483 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 = ∅) → (∅↑↑suc 𝑚) = ∅) |
18 | finxpsuc 36184 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ((∅↑↑𝑚) × ∅)) | |
19 | xp0 6149 | . . . . . 6 ⊢ ((∅↑↑𝑚) × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2789 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ∅) |
21 | 17, 20 | pm2.61dane 3030 | . . . 4 ⊢ (𝑚 ∈ ω → (∅↑↑suc 𝑚) = ∅) |
22 | 21 | a1d 25 | . . 3 ⊢ (𝑚 ∈ ω → ((∅↑↑𝑚) = ∅ → (∅↑↑suc 𝑚) = ∅)) |
23 | 2, 4, 6, 8, 9, 22 | finds 7876 | . 2 ⊢ (𝑁 ∈ ω → (∅↑↑𝑁) = ∅) |
24 | finxpnom 36187 | . 2 ⊢ (¬ 𝑁 ∈ ω → (∅↑↑𝑁) = ∅) | |
25 | 23, 24 | pm2.61i 182 | 1 ⊢ (∅↑↑𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∅c0 4320 × cxp 5670 suc csuc 6358 ωcom 7842 1oc1o 8446 ↑↑cfinxp 36169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-en 8928 df-fin 8931 df-finxp 36170 |
This theorem is referenced by: (None) |
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