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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 33224 | . . . 4 ⊢ (𝑛 = ∅ → (∅↑↑𝑛) = (∅↑↑∅)) | |
2 | 1 | eqeq1d 2624 | . . 3 ⊢ (𝑛 = ∅ → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑∅) = ∅)) |
3 | finxpeq2 33224 | . . . 4 ⊢ (𝑛 = 𝑚 → (∅↑↑𝑛) = (∅↑↑𝑚)) | |
4 | 3 | eqeq1d 2624 | . . 3 ⊢ (𝑛 = 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑚) = ∅)) |
5 | finxpeq2 33224 | . . . 4 ⊢ (𝑛 = suc 𝑚 → (∅↑↑𝑛) = (∅↑↑suc 𝑚)) | |
6 | 5 | eqeq1d 2624 | . . 3 ⊢ (𝑛 = suc 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑suc 𝑚) = ∅)) |
7 | finxpeq2 33224 | . . . 4 ⊢ (𝑛 = 𝑁 → (∅↑↑𝑛) = (∅↑↑𝑁)) | |
8 | 7 | eqeq1d 2624 | . . 3 ⊢ (𝑛 = 𝑁 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑁) = ∅)) |
9 | finxp0 33228 | . . 3 ⊢ (∅↑↑∅) = ∅ | |
10 | suceq 5790 | . . . . . . . . 9 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅) | |
11 | df-1o 7560 | . . . . . . . . 9 ⊢ 1𝑜 = suc ∅ | |
12 | 10, 11 | syl6eqr 2674 | . . . . . . . 8 ⊢ (𝑚 = ∅ → suc 𝑚 = 1𝑜) |
13 | finxpeq2 33224 | . . . . . . . 8 ⊢ (suc 𝑚 = 1𝑜 → (∅↑↑suc 𝑚) = (∅↑↑1𝑜)) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = (∅↑↑1𝑜)) |
15 | finxp1o 33229 | . . . . . . 7 ⊢ (∅↑↑1𝑜) = ∅ | |
16 | 14, 15 | syl6eq 2672 | . . . . . 6 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = ∅) |
17 | 16 | adantl 482 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 = ∅) → (∅↑↑suc 𝑚) = ∅) |
18 | finxpsuc 33235 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ((∅↑↑𝑚) × ∅)) | |
19 | xp0 5552 | . . . . . 6 ⊢ ((∅↑↑𝑚) × ∅) = ∅ | |
20 | 18, 19 | syl6eq 2672 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ∅) |
21 | 17, 20 | pm2.61dane 2881 | . . . 4 ⊢ (𝑚 ∈ ω → (∅↑↑suc 𝑚) = ∅) |
22 | 21 | a1d 25 | . . 3 ⊢ (𝑚 ∈ ω → ((∅↑↑𝑚) = ∅ → (∅↑↑suc 𝑚) = ∅)) |
23 | 2, 4, 6, 8, 9, 22 | finds 7092 | . 2 ⊢ (𝑁 ∈ ω → (∅↑↑𝑁) = ∅) |
24 | finxpnom 33238 | . 2 ⊢ (¬ 𝑁 ∈ ω → (∅↑↑𝑁) = ∅) | |
25 | 23, 24 | pm2.61i 176 | 1 ⊢ (∅↑↑𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 × cxp 5112 suc csuc 5725 ωcom 7065 1𝑜c1o 7553 ↑↑cfinxp 33220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-finxp 33221 |
This theorem is referenced by: (None) |
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