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Mirrors > Home > MPE Home > Th. List > oe0m | Structured version Visualization version GIF version |
Description: Value of zero raised to an ordinal. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oe0m | โข (๐ด โ On โ (โ โo ๐ด) = (1o โ ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6375 | . . 3 โข โ โ On | |
2 | oev 8464 | . . 3 โข ((โ โ On โง ๐ด โ On) โ (โ โo ๐ด) = if(โ = โ , (1o โ ๐ด), (rec((๐ฅ โ V โฆ (๐ฅ ยทo โ )), 1o)โ๐ด))) | |
3 | 1, 2 | mpan 689 | . 2 โข (๐ด โ On โ (โ โo ๐ด) = if(โ = โ , (1o โ ๐ด), (rec((๐ฅ โ V โฆ (๐ฅ ยทo โ )), 1o)โ๐ด))) |
4 | eqid 2733 | . . 3 โข โ = โ | |
5 | 4 | iftruei 4497 | . 2 โข if(โ = โ , (1o โ ๐ด), (rec((๐ฅ โ V โฆ (๐ฅ ยทo โ )), 1o)โ๐ด)) = (1o โ ๐ด) |
6 | 3, 5 | eqtrdi 2789 | 1 โข (๐ด โ On โ (โ โo ๐ด) = (1o โ ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 Vcvv 3447 โ cdif 3911 โ c0 4286 ifcif 4490 โฆ cmpt 5192 Oncon0 6321 โcfv 6500 (class class class)co 7361 reccrdg 8359 1oc1o 8409 ยทo comu 8414 โo coe 8415 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-suc 6327 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oexp 8422 |
This theorem is referenced by: oe0m0 8470 oe0m1 8471 cantnflem2 9634 oe0rif 41667 |
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