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Mirrors > Home > MPE Home > Th. List > axregnd | Structured version Visualization version Unicode version |
Description: A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) |
Ref | Expression |
---|---|
axregnd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axregndlem2 9053 |
. . . 4
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2 | nfnae 2162 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | nfnae 2162 |
. . . . . 6
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4 | 2, 3 | nfan 2021 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | nfnae 2162 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | nfnae 2162 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | nfan 2021 |
. . . . . . 7
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8 | nfcvf 2625 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | nfcrd 2608 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 9 | adantr 471 |
. . . . . . . 8
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11 | nfcvf 2625 |
. . . . . . . . . . 11
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12 | 11 | nfcrd 2608 |
. . . . . . . . . 10
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13 | 12 | nfnd 1994 |
. . . . . . . . 9
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14 | 13 | adantl 472 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 10, 14 | nfimd 2010 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | elequ1 1904 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | elequ1 1904 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | notbid 300 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 16, 18 | imbi12d 326 |
. . . . . . . 8
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20 | 19 | a1i 11 |
. . . . . . 7
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21 | 7, 15, 20 | cbvald 2128 |
. . . . . 6
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22 | 21 | anbi2d 715 |
. . . . 5
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23 | 4, 22 | exbid 1974 |
. . . 4
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24 | 1, 23 | syl5ib 227 |
. . 3
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25 | 24 | ex 440 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | axregndlem1 9052 |
. . 3
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27 | 26 | aecoms 2156 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 19.8a 1945 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | nfae 2160 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | elirrv 8137 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() | |
31 | elequ2 1911 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 30, 31 | mtbii 308 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32 | a1d 26 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33 | alimi 1694 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | anim2i 577 |
. . . . 5
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36 | 35 | expcom 441 |
. . . 4
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37 | 29, 36 | eximd 1970 |
. . 3
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38 | 28, 37 | syl5 33 |
. 2
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39 | 25, 27, 38 | pm2.61ii 170 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pr 4652 ax-reg 8132 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-v 3058 df-dif 3418 df-un 3420 df-nul 3743 df-sn 3980 df-pr 3982 |
This theorem is referenced by: zfcndreg 9067 axregprim 30380 |
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