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Mirrors > Home > ILE Home > Th. List > op1stb | Unicode version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
op1stb.1 | |
op1stb.2 |
Ref | Expression |
---|---|
op1stb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1stb.1 | . . . . . 6 | |
2 | op1stb.2 | . . . . . 6 | |
3 | 1, 2 | dfop 3548 | . . . . 5 |
4 | 3 | inteqi 3619 | . . . 4 |
5 | snexgOLD 3935 | . . . . . . 7 | |
6 | 1, 5 | ax-mp 7 | . . . . . 6 |
7 | prexgOLD 3946 | . . . . . . 7 | |
8 | 1, 2, 7 | mp2an 402 | . . . . . 6 |
9 | 6, 8 | intpr 3647 | . . . . 5 |
10 | snsspr1 3512 | . . . . . 6 | |
11 | df-ss 2931 | . . . . . 6 | |
12 | 10, 11 | mpbi 133 | . . . . 5 |
13 | 9, 12 | eqtri 2060 | . . . 4 |
14 | 4, 13 | eqtri 2060 | . . 3 |
15 | 14 | inteqi 3619 | . 2 |
16 | 1 | intsn 3650 | . 2 |
17 | 15, 16 | eqtri 2060 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1243 wcel 1393 cvv 2557 cin 2916 wss 2917 csn 3375 cpr 3376 cop 3378 cint 3615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-int 3616 |
This theorem is referenced by: elreldm 4560 op2ndb 4804 1stval2 5782 fundmen 6286 xpsnen 6295 |
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