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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr21val | Structured version Visualization version Unicode version |
Description: Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-pr21val | pr1 (|, |) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-2upl 32999 | . . 3 (|, |) (||) tag | |
2 | bj-pr1eq 32990 | . . 3 (|, |) (||) tag pr1 (|, |) pr1 (||) tag | |
3 | 1, 2 | ax-mp 5 | . 2 pr1 (|, |) pr1 (||) tag |
4 | bj-pr1un 32991 | . 2 pr1 (||) tag pr1 (||) pr1 tag | |
5 | bj-pr11val 32993 | . . . 4 pr1 (||) | |
6 | bj-pr1val 32992 | . . . . 5 pr1 tag | |
7 | 1n0 7575 | . . . . . . 7 | |
8 | 7 | neii 2796 | . . . . . 6 |
9 | 8 | iffalsei 4096 | . . . . 5 |
10 | 6, 9 | eqtri 2644 | . . . 4 pr1 tag |
11 | 5, 10 | uneq12i 3765 | . . 3 pr1 (||) pr1 tag |
12 | un0 3967 | . . 3 | |
13 | 11, 12 | eqtri 2644 | . 2 pr1 (||) pr1 tag |
14 | 3, 4, 13 | 3eqtri 2648 | 1 pr1 (|, |) |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 cun 3572 c0 3915 cif 4086 csn 4177 cxp 5112 c1o 7553 tag bj-ctag 32962 (|bj-c1upl 32985 pr1 bj-cpr1 32988 (|bj-c2uple 32998 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 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-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-1o 7560 df-bj-sngl 32954 df-bj-tag 32963 df-bj-proj 32979 df-bj-1upl 32986 df-bj-pr1 32989 df-bj-2upl 32999 |
This theorem is referenced by: bj-2uplth 33009 bj-2uplex 33010 |
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