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Mirrors > Home > MPE Home > Th. List > om0 | Structured version Unicode version |
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
om0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 4883 |
. . 3
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2 | omv 7065 |
. . 3
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3 | 1, 2 | mpan2 671 |
. 2
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4 | 0ex 4533 |
. . 3
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5 | 4 | rdg0 6990 |
. 2
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6 | 3, 5 | syl6eq 2511 |
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 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
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 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-recs 6945 df-rdg 6979 df-omul 7038 |
This theorem is referenced by: om0x 7072 oesuclem 7078 omcl 7089 om1 7094 omwordri 7124 om00 7127 odi 7131 omass 7132 oen0 7138 oeoa 7149 oeoelem 7150 oeeui 7154 nnm0 7157 cantnfle 7993 cantnfp1 8003 cantnfleOLD 8023 cantnfp1OLD 8029 |
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