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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axregndlem2 9425 | . . . 4 | |
2 | nfnae 2318 | . . . . . 6 | |
3 | nfnae 2318 | . . . . . 6 | |
4 | 2, 3 | nfan 1828 | . . . . 5 |
5 | nfnae 2318 | . . . . . . . 8 | |
6 | nfnae 2318 | . . . . . . . 8 | |
7 | 5, 6 | nfan 1828 | . . . . . . 7 |
8 | nfcvf 2788 | . . . . . . . . . 10 | |
9 | 8 | nfcrd 2771 | . . . . . . . . 9 |
10 | 9 | adantr 481 | . . . . . . . 8 |
11 | nfcvf 2788 | . . . . . . . . . . 11 | |
12 | 11 | nfcrd 2771 | . . . . . . . . . 10 |
13 | 12 | nfnd 1785 | . . . . . . . . 9 |
14 | 13 | adantl 482 | . . . . . . . 8 |
15 | 10, 14 | nfimd 1823 | . . . . . . 7 |
16 | elequ1 1997 | . . . . . . . . 9 | |
17 | elequ1 1997 | . . . . . . . . . 10 | |
18 | 17 | notbid 308 | . . . . . . . . 9 |
19 | 16, 18 | imbi12d 334 | . . . . . . . 8 |
20 | 19 | a1i 11 | . . . . . . 7 |
21 | 7, 15, 20 | cbvald 2277 | . . . . . 6 |
22 | 21 | anbi2d 740 | . . . . 5 |
23 | 4, 22 | exbid 2091 | . . . 4 |
24 | 1, 23 | syl5ib 234 | . . 3 |
25 | 24 | ex 450 | . 2 |
26 | axregndlem1 9424 | . . 3 | |
27 | 26 | aecoms 2312 | . 2 |
28 | 19.8a 2052 | . . 3 | |
29 | nfae 2316 | . . . 4 | |
30 | elirrv 8504 | . . . . . . . . 9 | |
31 | elequ2 2004 | . . . . . . . . 9 | |
32 | 30, 31 | mtbii 316 | . . . . . . . 8 |
33 | 32 | a1d 25 | . . . . . . 7 |
34 | 33 | alimi 1739 | . . . . . 6 |
35 | 34 | anim2i 593 | . . . . 5 |
36 | 35 | expcom 451 | . . . 4 |
37 | 29, 36 | eximd 2085 | . . 3 |
38 | 28, 37 | syl5 34 | . 2 |
39 | 25, 27, 38 | pm2.61ii 177 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wal 1481 wex 1704 wnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: zfcndreg 9439 axregprim 31582 |
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