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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 32400 | . . . 4 ⊢ (𝑛 = ∅ → (∅↑↑𝑛) = (∅↑↑∅)) | |
2 | 1 | eqeq1d 2612 | . . 3 ⊢ (𝑛 = ∅ → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑∅) = ∅)) |
3 | finxpeq2 32400 | . . . 4 ⊢ (𝑛 = 𝑚 → (∅↑↑𝑛) = (∅↑↑𝑚)) | |
4 | 3 | eqeq1d 2612 | . . 3 ⊢ (𝑛 = 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑚) = ∅)) |
5 | finxpeq2 32400 | . . . 4 ⊢ (𝑛 = suc 𝑚 → (∅↑↑𝑛) = (∅↑↑suc 𝑚)) | |
6 | 5 | eqeq1d 2612 | . . 3 ⊢ (𝑛 = suc 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑suc 𝑚) = ∅)) |
7 | finxpeq2 32400 | . . . 4 ⊢ (𝑛 = 𝑁 → (∅↑↑𝑛) = (∅↑↑𝑁)) | |
8 | 7 | eqeq1d 2612 | . . 3 ⊢ (𝑛 = 𝑁 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑁) = ∅)) |
9 | finxp0 32404 | . . 3 ⊢ (∅↑↑∅) = ∅ | |
10 | suceq 5707 | . . . . . . . . 9 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅) | |
11 | df-1o 7447 | . . . . . . . . 9 ⊢ 1𝑜 = suc ∅ | |
12 | 10, 11 | syl6eqr 2662 | . . . . . . . 8 ⊢ (𝑚 = ∅ → suc 𝑚 = 1𝑜) |
13 | finxpeq2 32400 | . . . . . . . 8 ⊢ (suc 𝑚 = 1𝑜 → (∅↑↑suc 𝑚) = (∅↑↑1𝑜)) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = (∅↑↑1𝑜)) |
15 | finxp1o 32405 | . . . . . . 7 ⊢ (∅↑↑1𝑜) = ∅ | |
16 | 14, 15 | syl6eq 2660 | . . . . . 6 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = ∅) |
17 | 16 | adantl 481 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 = ∅) → (∅↑↑suc 𝑚) = ∅) |
18 | finxpsuc 32411 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ((∅↑↑𝑚) × ∅)) | |
19 | xp0 5471 | . . . . . 6 ⊢ ((∅↑↑𝑚) × ∅) = ∅ | |
20 | 18, 19 | syl6eq 2660 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ∅) |
21 | 17, 20 | pm2.61dane 2869 | . . . 4 ⊢ (𝑚 ∈ ω → (∅↑↑suc 𝑚) = ∅) |
22 | 21 | a1d 25 | . . 3 ⊢ (𝑚 ∈ ω → ((∅↑↑𝑚) = ∅ → (∅↑↑suc 𝑚) = ∅)) |
23 | 2, 4, 6, 8, 9, 22 | finds 6984 | . 2 ⊢ (𝑁 ∈ ω → (∅↑↑𝑁) = ∅) |
24 | finxpnom 32414 | . 2 ⊢ (¬ 𝑁 ∈ ω → (∅↑↑𝑁) = ∅) | |
25 | 23, 24 | pm2.61i 175 | 1 ⊢ (∅↑↑𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 × cxp 5036 suc csuc 5642 ωcom 6957 1𝑜c1o 7440 ↑↑cfinxp 32396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-finxp 32397 |
This theorem is referenced by: (None) |
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