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Mirrors > Home > MPE Home > Th. List > axregndlem1 | Structured version Visualization version Unicode version |
Description: Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) |
Ref | Expression |
---|---|
axregndlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2052 | . 2 | |
2 | nfae 2316 | . . 3 | |
3 | nfae 2316 | . . . . . 6 | |
4 | elirrv 8504 | . . . . . . . . 9 | |
5 | elequ1 1997 | . . . . . . . . 9 | |
6 | 4, 5 | mtbii 316 | . . . . . . . 8 |
7 | 6 | sps 2055 | . . . . . . 7 |
8 | 7 | pm2.21d 118 | . . . . . 6 |
9 | 3, 8 | alrimi 2082 | . . . . 5 |
10 | 9 | anim2i 593 | . . . 4 |
11 | 10 | expcom 451 | . . 3 |
12 | 2, 11 | eximd 2085 | . 2 |
13 | 1, 12 | syl5 34 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wal 1481 wex 1704 |
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: axregndlem2 9425 axregnd 9426 |
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