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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 37370 | . . . 4 ⊢ (𝑛 = ∅ → (∅↑↑𝑛) = (∅↑↑∅)) | |
2 | 1 | eqeq1d 2737 | . . 3 ⊢ (𝑛 = ∅ → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑∅) = ∅)) |
3 | finxpeq2 37370 | . . . 4 ⊢ (𝑛 = 𝑚 → (∅↑↑𝑛) = (∅↑↑𝑚)) | |
4 | 3 | eqeq1d 2737 | . . 3 ⊢ (𝑛 = 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑚) = ∅)) |
5 | finxpeq2 37370 | . . . 4 ⊢ (𝑛 = suc 𝑚 → (∅↑↑𝑛) = (∅↑↑suc 𝑚)) | |
6 | 5 | eqeq1d 2737 | . . 3 ⊢ (𝑛 = suc 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑suc 𝑚) = ∅)) |
7 | finxpeq2 37370 | . . . 4 ⊢ (𝑛 = 𝑁 → (∅↑↑𝑛) = (∅↑↑𝑁)) | |
8 | 7 | eqeq1d 2737 | . . 3 ⊢ (𝑛 = 𝑁 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑁) = ∅)) |
9 | finxp0 37374 | . . 3 ⊢ (∅↑↑∅) = ∅ | |
10 | suceq 6452 | . . . . . . . . 9 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅) | |
11 | df-1o 8505 | . . . . . . . . 9 ⊢ 1o = suc ∅ | |
12 | 10, 11 | eqtr4di 2793 | . . . . . . . 8 ⊢ (𝑚 = ∅ → suc 𝑚 = 1o) |
13 | finxpeq2 37370 | . . . . . . . 8 ⊢ (suc 𝑚 = 1o → (∅↑↑suc 𝑚) = (∅↑↑1o)) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = (∅↑↑1o)) |
15 | finxp1o 37375 | . . . . . . 7 ⊢ (∅↑↑1o) = ∅ | |
16 | 14, 15 | eqtrdi 2791 | . . . . . 6 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = ∅) |
17 | 16 | adantl 481 | . . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 = ∅) → (∅↑↑suc 𝑚) = ∅) |
18 | finxpsuc 37381 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ((∅↑↑𝑚) × ∅)) | |
19 | xp0 6180 | . . . . . 6 ⊢ ((∅↑↑𝑚) × ∅) = ∅ | |
20 | 18, 19 | eqtrdi 2791 | . . . . 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 7919 | . 2 ⊢ (𝑁 ∈ ω → (∅↑↑𝑁) = ∅) |
24 | finxpnom 37384 | . 2 ⊢ (¬ 𝑁 ∈ ω → (∅↑↑𝑁) = ∅) | |
25 | 23, 24 | pm2.61i 182 | 1 ⊢ (∅↑↑𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 × cxp 5687 suc csuc 6388 ωcom 7887 1oc1o 8498 ↑↑cfinxp 37366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-en 8985 df-fin 8988 df-finxp 37367 |
This theorem is referenced by: (None) |
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