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| Mirrors > Home > NFE Home > Th. List > intirr | Unicode version | ||
| Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Revised by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| intirr |
            |
,
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | incom 3448 |
. . . 4
| |
| 2 | 1 | eqeq1i 2360 |
. . 3
|
| 3 | disj5 3890 |
. . 3
 ∼ | |
| 4 | ssrel 4844 |
. . 3
∼     
∼ | |
| 5 | 2, 3, 4 | 3bitri 262 |
. 2
∼ |
| 6 | vex 2862 |
. . . . . 6
 | |
| 7 | 6 | ideq 4870 |
. . . . 5
|
| 8 | df-br 4640 |
. . . . 5
| |
| 9 | 7, 8 | bitr3i 242 |
. . . 4
|
| 10 | df-br 4640 |
. . . . . 6
| |
| 11 | 10 | notbii 287 |
. . . . 5
|
| 12 | vex 2862 |
. . . . . . 7
| |
| 13 | 12, 6 | opex 4588 |
. . . . . 6
|
| 14 | 13 | elcompl 3225 |
. . . . 5
∼ |
| 15 | 11, 14 | bitr4i 243 |
. . . 4
∼ |
| 16 | 9, 15 | imbi12i 316 |
. . 3
∼ |
| 17 | 16 | 2albii 1567 |
. 2
∼ |
| 18 | equcom 1680 |
. . . . . 6
| |
| 19 | 18 | imbi1i 315 |
. . . . 5
|
| 20 | 19 | albii 1566 |
. . . 4
|
| 21 | breq2 4643 |
. . . . . 6
| |
| 22 | 21 | notbid 285 |
. . . . 5
|
| 23 | 12, 22 | ceqsalv 2885 |
. . . 4
|
| 24 | 20, 23 | bitri 240 |
. . 3
|
| 25 | 24 | albii 1566 |
. 2
|
| 26 | 5, 17, 25 | 3bitr2i 264 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wb 176
wal 1540
wceq 1642
wcel 1710
∼ ccompl 3205 cin 3208 wss 3257 c0 3550 cop 4561 class class class wbr 4639
cid 4763 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
| This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-id 4767 |
| This theorem is referenced by: (None) |
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