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Mirrors > Home > MPE Home > Th. List > op1stb | Structured version Visualization version Unicode version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5619 to extract the second member, op1sta 5617 for an alternate version, and op1st 7176 for the preferred version.) (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 4401 | . . . . 5 |
4 | 3 | inteqi 4479 | . . . 4 |
5 | snex 4908 | . . . . . 6 | |
6 | prex 4909 | . . . . . 6 | |
7 | 5, 6 | intpr 4510 | . . . . 5 |
8 | snsspr1 4345 | . . . . . 6 | |
9 | df-ss 3588 | . . . . . 6 | |
10 | 8, 9 | mpbi 220 | . . . . 5 |
11 | 7, 10 | eqtri 2644 | . . . 4 |
12 | 4, 11 | eqtri 2644 | . . 3 |
13 | 12 | inteqi 4479 | . 2 |
14 | 1 | intsn 4513 | . 2 |
15 | 13, 14 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 cin 3573 wss 3574 csn 4177 cpr 4179 cop 4183 cint 4475 |
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-3an 1039 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-ral 2917 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-int 4476 |
This theorem is referenced by: elreldm 5350 op2ndb 5619 elxp5 7111 1stval2 7185 fundmen 8030 xpsnen 8044 |
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