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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3791 |
. . . . 5
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2 | 1 | inteqd 3864 |
. . . 4
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3 | snexg 4202 |
. . . . . 6
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4 | prexg 4229 |
. . . . . 6
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5 | intprg 3892 |
. . . . . 6
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6 | 3, 4, 5 | syl2an2r 595 |
. . . . 5
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7 | snsspr1 3755 |
. . . . . 6
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8 | df-ss 3157 |
. . . . . 6
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9 | 7, 8 | mpbi 145 |
. . . . 5
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10 | 6, 9 | eqtrdi 2238 |
. . . 4
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11 | 2, 10 | eqtrd 2222 |
. . 3
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12 | 11 | inteqd 3864 |
. 2
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13 | intsng 3893 |
. . 3
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14 | 13 | adantr 276 |
. 2
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15 | 12, 14 | eqtrd 2222 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-int 3860 |
This theorem is referenced by: elxp5 5135 fundmen 6833 |
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