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Mirrors > Home > ILE Home > Th. List > disj | Unicode version |
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
disj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-in 3150 |
. . . 4
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2 | 1 | eqeq1i 2197 |
. . 3
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3 | abeq1 2299 |
. . 3
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4 | imnan 691 |
. . . . 5
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5 | noel 3441 |
. . . . . 6
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6 | 5 | nbn 700 |
. . . . 5
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7 | 4, 6 | bitr2i 185 |
. . . 4
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8 | 7 | albii 1481 |
. . 3
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9 | 2, 3, 8 | 3bitri 206 |
. 2
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10 | df-ral 2473 |
. 2
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11 | 9, 10 | bitr4i 187 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-in 3150 df-nul 3438 |
This theorem is referenced by: disjr 3487 disj1 3488 disjne 3491 f0rn0 5426 renfdisj 8042 fvinim0ffz 10266 fxnn0nninf 10464 fprodsplitdc 11631 exmidunben 12472 dedekindeulemuub 14532 dedekindeulemlu 14536 dedekindicclemuub 14541 dedekindicclemlu 14545 ivthinclemdisj 14555 exmidsbthrlem 15208 |
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