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Mirrors > Home > ILE Home > Th. List > relopabi | Unicode version |
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
relopabi.1 |
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Ref | Expression |
---|---|
relopabi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopabi.1 |
. . . 4
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2 | df-opab 4077 |
. . . 4
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3 | 1, 2 | eqtri 2208 |
. . 3
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4 | vex 2752 |
. . . . . . . 8
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5 | vex 2752 |
. . . . . . . 8
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6 | 4, 5 | opelvv 4688 |
. . . . . . 7
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7 | eleq1 2250 |
. . . . . . 7
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8 | 6, 7 | mpbiri 168 |
. . . . . 6
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9 | 8 | adantr 276 |
. . . . 5
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10 | 9 | exlimivv 1906 |
. . . 4
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11 | 10 | abssi 3242 |
. . 3
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12 | 3, 11 | eqsstri 3199 |
. 2
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13 | df-rel 4645 |
. 2
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14 | 12, 13 | mpbir 146 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 df-xp 4644 df-rel 4645 |
This theorem is referenced by: relopab 4765 mptrel 4767 reli 4768 rele 4769 relcnv 5018 cotr 5022 relco 5139 reloprab 5936 reldmoprab 5973 eqer 6580 ecopover 6646 ecopoverg 6649 relen 6757 reldom 6758 enq0er 7447 aprcl 8616 aptap 8620 climrel 11301 brstruct 12484 |
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