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| Description: Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) | 
| Ref | Expression | 
|---|---|
| oe0 | ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq1 7438 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o ∅) = (∅ ↑o ∅)) | |
| 2 | oe0m0 8558 | . . . . 5 ⊢ (∅ ↑o ∅) = 1o | |
| 3 | 1, 2 | eqtrdi 2793 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑o ∅) = 1o) | 
| 4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 = ∅) → (𝐴 ↑o ∅) = 1o) | 
| 5 | 0elon 6438 | . . . . . 6 ⊢ ∅ ∈ On | |
| 6 | oevn0 8553 | . . . . . 6 ⊢ (((𝐴 ∈ On ∧ ∅ ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅)) | |
| 7 | 5, 6 | mpanl2 701 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅)) | 
| 8 | 1oex 8516 | . . . . . 6 ⊢ 1o ∈ V | |
| 9 | 8 | rdg0 8461 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘∅) = 1o | 
| 10 | 7, 9 | eqtrdi 2793 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = 1o) | 
| 11 | 10 | adantll 714 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o ∅) = 1o) | 
| 12 | 4, 11 | oe0lem 8551 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑o ∅) = 1o) | 
| 13 | 12 | anidms 566 | 1 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 Oncon0 6384 ‘cfv 6561 (class class class)co 7431 reccrdg 8449 1oc1o 8499 ·o comu 8504 ↑o coe 8505 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oexp 8512 | 
| This theorem is referenced by: oecl 8575 oe1 8582 oe1m 8583 oen0 8624 oewordri 8630 oeoalem 8634 oeoelem 8636 oeoe 8637 oeeulem 8639 nnecl 8651 oaabs2 8687 cantnff 9714 onexoegt 43256 oe0suclim 43290 oenassex 43331 omabs2 43345 omcl2 43346 | 
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