Theorem finxp00 34686

Description: Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)

Assertion

Ref	Expression
finxp00	⊢ (∅↑↑𝑁) = ∅

Proof of Theorem finxp00

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		finxpeq2 34671	. . . 4 ⊢ (𝑛 = ∅ → (∅↑↑𝑛) = (∅↑↑∅))
2	1	eqeq1d 2823	. . 3 ⊢ (𝑛 = ∅ → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑∅) = ∅))
3		finxpeq2 34671	. . . 4 ⊢ (𝑛 = 𝑚 → (∅↑↑𝑛) = (∅↑↑𝑚))
4	3	eqeq1d 2823	. . 3 ⊢ (𝑛 = 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑚) = ∅))
5		finxpeq2 34671	. . . 4 ⊢ (𝑛 = suc 𝑚 → (∅↑↑𝑛) = (∅↑↑suc 𝑚))
6	5	eqeq1d 2823	. . 3 ⊢ (𝑛 = suc 𝑚 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑suc 𝑚) = ∅))
7		finxpeq2 34671	. . . 4 ⊢ (𝑛 = 𝑁 → (∅↑↑𝑛) = (∅↑↑𝑁))
8	7	eqeq1d 2823	. . 3 ⊢ (𝑛 = 𝑁 → ((∅↑↑𝑛) = ∅ ↔ (∅↑↑𝑁) = ∅))
9		finxp0 34675	. . 3 ⊢ (∅↑↑∅) = ∅
10		suceq 6256	. . . . . . . . 9 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅)
11		df-1o 8102	. . . . . . . . 9 ⊢ 1o = suc ∅
12	10, 11	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑚 = ∅ → suc 𝑚 = 1o)
13		finxpeq2 34671	. . . . . . . 8 ⊢ (suc 𝑚 = 1o → (∅↑↑suc 𝑚) = (∅↑↑1o))
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = (∅↑↑1o))
15		finxp1o 34676	. . . . . . 7 ⊢ (∅↑↑1o) = ∅
16	14, 15	syl6eq 2872	. . . . . 6 ⊢ (𝑚 = ∅ → (∅↑↑suc 𝑚) = ∅)
17	16	adantl 484	. . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 = ∅) → (∅↑↑suc 𝑚) = ∅)
18		finxpsuc 34682	. . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ((∅↑↑𝑚) × ∅))
19		xp0 6015	. . . . . 6 ⊢ ((∅↑↑𝑚) × ∅) = ∅
20	18, 19	syl6eq 2872	. . . . 5 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → (∅↑↑suc 𝑚) = ∅)
21	17, 20	pm2.61dane 3104	. . . 4 ⊢ (𝑚 ∈ ω → (∅↑↑suc 𝑚) = ∅)
22	21	a1d 25	. . 3 ⊢ (𝑚 ∈ ω → ((∅↑↑𝑚) = ∅ → (∅↑↑suc 𝑚) = ∅))
23	2, 4, 6, 8, 9, 22	finds 7608	. 2 ⊢ (𝑁 ∈ ω → (∅↑↑𝑁) = ∅)
24		finxpnom 34685	. 2 ⊢ (¬ 𝑁 ∈ ω → (∅↑↑𝑁) = ∅)
25	23, 24	pm2.61i 184	1 ⊢ (∅↑↑𝑁) = ∅

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∅c0 4291   × cxp 5553  suc csuc 6193  ωcom 7580  1_oc1o 8095  ↑↑cfinxp 34667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-finxp 34668

This theorem is referenced by: (None)