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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 3750 | . . . . 5 | |
2 | 1 | inteqd 3823 | . . . 4 |
3 | snexg 4157 | . . . . . 6 | |
4 | prexg 4183 | . . . . . 6 | |
5 | intprg 3851 | . . . . . 6 | |
6 | 3, 4, 5 | syl2an2r 585 | . . . . 5 |
7 | snsspr1 3715 | . . . . . 6 | |
8 | df-ss 3124 | . . . . . 6 | |
9 | 7, 8 | mpbi 144 | . . . . 5 |
10 | 6, 9 | eqtrdi 2213 | . . . 4 |
11 | 2, 10 | eqtrd 2197 | . . 3 |
12 | 11 | inteqd 3823 | . 2 |
13 | intsng 3852 | . . 3 | |
14 | 13 | adantr 274 | . 2 |
15 | 12, 14 | eqtrd 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cvv 2721 cin 3110 wss 3111 csn 3570 cpr 3571 cop 3573 cint 3818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-int 3819 |
This theorem is referenced by: elxp5 5086 fundmen 6763 |
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