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Mirrors > Home > MPE Home > Th. List > oevn0 | Structured version Unicode version |
Description: Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
oevn0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on0eln0 4877 |
. . . . 5
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2 | df-ne 2647 |
. . . . 5
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3 | 1, 2 | syl6bb 261 |
. . . 4
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4 | 3 | adantr 465 |
. . 3
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5 | oev 7059 |
. . . . 5
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6 | iffalse 3902 |
. . . . 5
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7 | 5, 6 | sylan9eq 2513 |
. . . 4
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8 | 7 | ex 434 |
. . 3
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9 | 4, 8 | sylbid 215 |
. 2
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10 | 9 | imp 429 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-rab 2805 df-v 3074 df-sbc 3289 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-iota 5484 df-fun 5523 df-fv 5529 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-recs 6937 df-rdg 6971 df-1o 7025 df-oexp 7031 |
This theorem is referenced by: oe0 7067 oev2 7068 oesuclem 7070 oelim 7079 |
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