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Mirrors > Home > MPE Home > Th. List > opthwiener | Structured version Visualization version Unicode version |
Description: Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations," Proc. of the Cambridge Philos. Soc., 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e. are not proper classes). See df-op 4184 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.) |
Ref | Expression |
---|---|
opthw.1 | |
opthw.2 |
Ref | Expression |
---|---|
opthwiener |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . 7 | |
2 | snex 4908 | . . . . . . . . . . . 12 | |
3 | 2 | prid2 4298 | . . . . . . . . . . 11 |
4 | eleq2 2690 | . . . . . . . . . . 11 | |
5 | 3, 4 | mpbii 223 | . . . . . . . . . 10 |
6 | 2 | elpr 4198 | . . . . . . . . . 10 |
7 | 5, 6 | sylib 208 | . . . . . . . . 9 |
8 | 0ex 4790 | . . . . . . . . . . . . 13 | |
9 | 8 | prid2 4298 | . . . . . . . . . . . 12 |
10 | opthw.2 | . . . . . . . . . . . . . 14 | |
11 | 10 | snnz 4309 | . . . . . . . . . . . . 13 |
12 | 8 | elsn 4192 | . . . . . . . . . . . . . 14 |
13 | eqcom 2629 | . . . . . . . . . . . . . 14 | |
14 | 12, 13 | bitri 264 | . . . . . . . . . . . . 13 |
15 | 11, 14 | nemtbir 2889 | . . . . . . . . . . . 12 |
16 | nelneq2 2726 | . . . . . . . . . . . 12 | |
17 | 9, 15, 16 | mp2an 708 | . . . . . . . . . . 11 |
18 | eqcom 2629 | . . . . . . . . . . 11 | |
19 | 17, 18 | mtbi 312 | . . . . . . . . . 10 |
20 | biorf 420 | . . . . . . . . . 10 | |
21 | 19, 20 | ax-mp 5 | . . . . . . . . 9 |
22 | 7, 21 | sylibr 224 | . . . . . . . 8 |
23 | 22 | preq2d 4275 | . . . . . . 7 |
24 | 1, 23 | eqtr4d 2659 | . . . . . 6 |
25 | prex 4909 | . . . . . . 7 | |
26 | prex 4909 | . . . . . . 7 | |
27 | 25, 26 | preqr1 4379 | . . . . . 6 |
28 | 24, 27 | syl 17 | . . . . 5 |
29 | snex 4908 | . . . . . 6 | |
30 | snex 4908 | . . . . . 6 | |
31 | 29, 30 | preqr1 4379 | . . . . 5 |
32 | 28, 31 | syl 17 | . . . 4 |
33 | opthw.1 | . . . . 5 | |
34 | 33 | sneqr 4371 | . . . 4 |
35 | 32, 34 | syl 17 | . . 3 |
36 | snex 4908 | . . . . . 6 | |
37 | 36 | sneqr 4371 | . . . . 5 |
38 | 22, 37 | syl 17 | . . . 4 |
39 | 10 | sneqr 4371 | . . . 4 |
40 | 38, 39 | syl 17 | . . 3 |
41 | 35, 40 | jca 554 | . 2 |
42 | sneq 4187 | . . . . 5 | |
43 | 42 | preq1d 4274 | . . . 4 |
44 | 43 | preq1d 4274 | . . 3 |
45 | sneq 4187 | . . . . 5 | |
46 | sneq 4187 | . . . . 5 | |
47 | 45, 46 | syl 17 | . . . 4 |
48 | 47 | preq2d 4275 | . . 3 |
49 | 44, 48 | sylan9eq 2676 | . 2 |
50 | 41, 49 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cvv 3200 c0 3915 csn 4177 cpr 4179 |
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-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 |
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-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: (None) |
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