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Mirrors > Home > MPE Home > Th. List > oe0m | Structured version Visualization version GIF version |
Description: Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oe0m | ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6246 | . . 3 ⊢ ∅ ∈ On | |
2 | oev 8141 | . . 3 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴))) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴))) |
4 | eqid 2823 | . . 3 ⊢ ∅ = ∅ | |
5 | 4 | iftruei 4476 | . 2 ⊢ if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o ∖ 𝐴) |
6 | 3, 5 | syl6eq 2874 | 1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∅c0 4293 ifcif 4469 ↦ cmpt 5148 Oncon0 6193 ‘cfv 6357 (class class class)co 7158 reccrdg 8047 1oc1o 8097 ·o comu 8102 ↑o coe 8103 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oexp 8110 |
This theorem is referenced by: oe0m0 8147 oe0m1 8148 cantnflem2 9155 |
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