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Mirrors > Home > ILE Home > Th. List > op1stbg | Unicode version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) |
Ref | Expression |
---|---|
op1stbg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3568 | . . . . 5 | |
2 | 1 | inteqd 3641 | . . . 4 |
3 | snexg 3956 | . . . . . 6 | |
4 | prexg 3966 | . . . . . 6 | |
5 | intprg 3669 | . . . . . 6 | |
6 | 3, 4, 5 | syl2an2r 559 | . . . . 5 |
7 | snsspr1 3533 | . . . . . 6 | |
8 | df-ss 2986 | . . . . . 6 | |
9 | 7, 8 | mpbi 143 | . . . . 5 |
10 | 6, 9 | syl6eq 2129 | . . . 4 |
11 | 2, 10 | eqtrd 2113 | . . 3 |
12 | 11 | inteqd 3641 | . 2 |
13 | intsng 3670 | . . 3 | |
14 | 13 | adantr 270 | . 2 |
15 | 12, 14 | eqtrd 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 cin 2972 wss 2973 csn 3398 cpr 3399 cop 3401 cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-int 3637 |
This theorem is referenced by: elxp5 4829 fundmen 6309 |
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