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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 6215 | . . 3 ⊢ ∅ ∈ On | |
2 | oev 8142 | . . 3 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴))) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴))) |
4 | eqid 2759 | . . 3 ⊢ ∅ = ∅ | |
5 | 4 | iftruei 4420 | . 2 ⊢ if(∅ = ∅, (1o ∖ 𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·o ∅)), 1o)‘𝐴)) = (1o ∖ 𝐴) |
6 | 3, 5 | eqtrdi 2810 | 1 ⊢ (𝐴 ∈ On → (∅ ↑o 𝐴) = (1o ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∖ cdif 3851 ∅c0 4221 ifcif 4413 ↦ cmpt 5105 Oncon0 6162 ‘cfv 6328 (class class class)co 7143 reccrdg 8048 1oc1o 8098 ·o comu 8103 ↑o coe 8104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-suc 6168 df-iota 6287 df-fun 6330 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oexp 8111 |
This theorem is referenced by: oe0m0 8148 oe0m1 8149 cantnflem2 9171 |
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