Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > 0neqopab | Structured version Visualization version Unicode version | ||
| Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
| Ref | Expression |
|---|---|
| 0neqopab |
             |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elopab 4983 |
. . 3
  "){kind=small} | |
| 2 | nfopab1 4719 |
. . . . . 6
| |
| 3 | 2 | nfel2 2781 |
. . . . 5
 |
| 4 | 3 | nfn 1784 |
. . . 4
|
| 5 | nfopab2 4720 |
. . . . . . 7
| |
| 6 | 5 | nfel2 2781 |
. . . . . 6
|
| 7 | 6 | nfn 1784 |
. . . . 5
|
| 8 | vex 3203 |
. . . . . . . 8
 | |
| 9 | vex 3203 |
. . . . . . . 8
| |
| 10 | 8, 9 | opnzi 4943 |
. . . . . . 7
 |
| 11 | nesym 2850 |
. . . . . . . 8
| |
| 12 | pm2.21 120 |
. . . . . . . 8
 | |
| 13 | 11, 12 | sylbi 207 |
. . . . . . 7
|
| 14 | 10, 13 | ax-mp 5 |
. . . . . 6
|
| 15 | 14 | adantr 481 |
. . . . 5
|
| 16 | 7, 15 | exlimi 2086 |
. . . 4
|
| 17 | 4, 16 | exlimi 2086 |
. . 3
|
| 18 | 1, 17 | sylbi 207 |
. 2
|
| 19 | id 22 |
. 2
| |
| 20 | 18, 19 | pm2.61i 176 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wa 384
wceq 1483
wex 1704
wcel 1990
wne 2794
c0 3915
cop 4183
copab 4712 |
| 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-ne 2795 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-opab 4713 |
| This theorem is referenced by: brabv 6699 bj-0nelmpt 33069 |
| Copyright terms: Public domain | W3C validator |