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Mirrors > Home > MPE Home > Th. List > oa0r | Structured version Visualization version Unicode version |
Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) |
Ref | Expression |
---|---|
oa0r |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6303 |
. . 3
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2 | id 22 |
. . 3
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3 | 1, 2 | eqeq12d 2468 |
. 2
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4 | oveq2 6303 |
. . 3
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5 | id 22 |
. . 3
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6 | 4, 5 | eqeq12d 2468 |
. 2
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7 | oveq2 6303 |
. . 3
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8 | id 22 |
. . 3
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9 | 7, 8 | eqeq12d 2468 |
. 2
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10 | oveq2 6303 |
. . 3
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11 | id 22 |
. . 3
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12 | 10, 11 | eqeq12d 2468 |
. 2
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13 | 0elon 5479 |
. . 3
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14 | oa0 7223 |
. . 3
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15 | 13, 14 | ax-mp 5 |
. 2
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16 | oasuc 7231 |
. . . . 5
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17 | 13, 16 | mpan 677 |
. . . 4
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18 | suceq 5491 |
. . . 4
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19 | 17, 18 | sylan9eq 2507 |
. . 3
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20 | 19 | ex 436 |
. 2
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21 | iuneq2 4298 |
. . . 4
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22 | uniiun 4334 |
. . . 4
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23 | 21, 22 | syl6eqr 2505 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | vex 3050 |
. . . . 5
![]() ![]() ![]() ![]() | |
25 | oalim 7239 |
. . . . . 6
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26 | 13, 25 | mpan 677 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 24, 26 | mpan 677 |
. . . 4
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28 | limuni 5486 |
. . . 4
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29 | 27, 28 | eqeq12d 2468 |
. . 3
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30 | 23, 29 | syl5ibr 225 |
. 2
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31 | 3, 6, 9, 12, 15, 20, 30 | tfinds 6691 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-oadd 7191 |
This theorem is referenced by: om1 7248 oaword2 7259 oeeui 7308 oaabs2 7351 cantnfp1 8191 |
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