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Mirrors > Home > MPE Home > Th. List > dfnn2 | Structured version Visualization version Unicode version |
Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 10643 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) |
Ref | Expression |
---|---|
dfnn2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 9669 |
. . . . 5
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2 | 1 | elintab 4259 |
. . . 4
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3 | simpl 463 |
. . . 4
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4 | 2, 3 | mpgbir 1684 |
. . 3
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5 | oveq1 6327 |
. . . . . . . . . 10
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6 | 5 | eleq1d 2524 |
. . . . . . . . 9
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7 | 6 | rspccv 3159 |
. . . . . . . 8
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8 | 7 | adantl 472 |
. . . . . . 7
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9 | 8 | a2i 14 |
. . . . . 6
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10 | 9 | alimi 1695 |
. . . . 5
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11 | vex 3060 |
. . . . . 6
![]() ![]() ![]() ![]() | |
12 | 11 | elintab 4259 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | ovex 6348 |
. . . . . 6
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14 | 13 | elintab 4259 |
. . . . 5
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15 | 10, 12, 14 | 3imtr4i 274 |
. . . 4
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16 | 15 | rgen 2759 |
. . 3
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17 | peano5nni 10645 |
. . 3
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18 | 4, 16, 17 | mp2an 683 |
. 2
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19 | 1nn 10653 |
. . . 4
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20 | peano2nn 10654 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | rgen 2759 |
. . . 4
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22 | nnex 10648 |
. . . . 5
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23 | eleq2 2529 |
. . . . . 6
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24 | eleq2 2529 |
. . . . . . 7
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25 | 24 | raleqbi1dv 3007 |
. . . . . 6
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26 | 23, 25 | anbi12d 722 |
. . . . 5
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27 | 22, 26 | elab 3197 |
. . . 4
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28 | 19, 21, 27 | mpbir2an 936 |
. . 3
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29 | intss1 4263 |
. . 3
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30 | 28, 29 | ax-mp 5 |
. 2
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31 | 18, 30 | eqssi 3460 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615 ax-cnex 9626 ax-resscn 9627 ax-1cn 9628 ax-icn 9629 ax-addcl 9630 ax-addrcl 9631 ax-mulcl 9632 ax-mulrcl 9633 ax-i2m1 9638 ax-1ne0 9639 ax-rrecex 9642 ax-cnre 9643 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-br 4419 df-opab 4478 df-mpt 4479 df-tr 4514 df-eprel 4767 df-id 4771 df-po 4777 df-so 4778 df-fr 4815 df-we 4817 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-pred 5403 df-ord 5449 df-on 5450 df-lim 5451 df-suc 5452 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-ov 6323 df-om 6725 df-wrecs 7059 df-recs 7121 df-rdg 7159 df-nn 10643 |
This theorem is referenced by: dfnn3 10656 |
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