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Mirrors > Home > MPE Home > Th. List > axinfndlem1 | Structured version Visualization version Unicode version |
Description: Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.) |
Ref | Expression |
---|---|
axinfndlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfinf 8536 | . . . . 5 | |
2 | nfnae 2318 | . . . . . . 7 | |
3 | nfnae 2318 | . . . . . . 7 | |
4 | 2, 3 | nfan 1828 | . . . . . 6 |
5 | nfcvf 2788 | . . . . . . . . 9 | |
6 | 5 | adantr 481 | . . . . . . . 8 |
7 | nfcvd 2765 | . . . . . . . 8 | |
8 | 6, 7 | nfeld 2773 | . . . . . . 7 |
9 | nfnae 2318 | . . . . . . . . 9 | |
10 | nfnae 2318 | . . . . . . . . 9 | |
11 | 9, 10 | nfan 1828 | . . . . . . . 8 |
12 | nfnae 2318 | . . . . . . . . . . 11 | |
13 | nfnae 2318 | . . . . . . . . . . 11 | |
14 | 12, 13 | nfan 1828 | . . . . . . . . . 10 |
15 | nfcvf 2788 | . . . . . . . . . . . . 13 | |
16 | 15 | adantl 482 | . . . . . . . . . . . 12 |
17 | 6, 16 | nfeld 2773 | . . . . . . . . . . 11 |
18 | 16, 7 | nfeld 2773 | . . . . . . . . . . 11 |
19 | 17, 18 | nfand 1826 | . . . . . . . . . 10 |
20 | 14, 19 | nfexd 2167 | . . . . . . . . 9 |
21 | 8, 20 | nfimd 1823 | . . . . . . . 8 |
22 | 11, 21 | nfald 2165 | . . . . . . 7 |
23 | 8, 22 | nfand 1826 | . . . . . 6 |
24 | simpr 477 | . . . . . . . . 9 | |
25 | 24 | eleq2d 2687 | . . . . . . . 8 |
26 | nfcvd 2765 | . . . . . . . . . . 11 | |
27 | nfcvf2 2789 | . . . . . . . . . . . 12 | |
28 | 27 | adantr 481 | . . . . . . . . . . 11 |
29 | 26, 28 | nfeqd 2772 | . . . . . . . . . 10 |
30 | 11, 29 | nfan1 2068 | . . . . . . . . 9 |
31 | nfcvd 2765 | . . . . . . . . . . . . 13 | |
32 | nfcvf2 2789 | . . . . . . . . . . . . . 14 | |
33 | 32 | adantl 482 | . . . . . . . . . . . . 13 |
34 | 31, 33 | nfeqd 2772 | . . . . . . . . . . . 12 |
35 | 14, 34 | nfan1 2068 | . . . . . . . . . . 11 |
36 | elequ2 2004 | . . . . . . . . . . . . 13 | |
37 | 36 | anbi2d 740 | . . . . . . . . . . . 12 |
38 | 37 | adantl 482 | . . . . . . . . . . 11 |
39 | 35, 38 | exbid 2091 | . . . . . . . . . 10 |
40 | 25, 39 | imbi12d 334 | . . . . . . . . 9 |
41 | 30, 40 | albid 2090 | . . . . . . . 8 |
42 | 25, 41 | anbi12d 747 | . . . . . . 7 |
43 | 42 | ex 450 | . . . . . 6 |
44 | 4, 23, 43 | cbvexd 2278 | . . . . 5 |
45 | 1, 44 | mpbii 223 | . . . 4 |
46 | 45 | a1d 25 | . . 3 |
47 | 46 | ex 450 | . 2 |
48 | nd1 9409 | . . 3 | |
49 | 48 | pm2.21d 118 | . 2 |
50 | nd2 9410 | . . 3 | |
51 | 50 | pm2.21d 118 | . 2 |
52 | 47, 49, 51 | pm2.61ii 177 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wal 1481 wex 1704 wnfc 2751 |
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 ax-inf 8535 |
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: axinfnd 9428 |
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