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Mirrors > Home > ILE Home > Th. List > unisucg | Unicode version |
Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.) |
Ref | Expression |
---|---|
unisucg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4251 |
. . . . . 6
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2 | 1 | unieqi 3710 |
. . . . 5
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3 | uniun 3719 |
. . . . 5
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4 | 2, 3 | eqtri 2133 |
. . . 4
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5 | unisng 3717 |
. . . . 5
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6 | 5 | uneq2d 3194 |
. . . 4
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7 | 4, 6 | syl5eq 2157 |
. . 3
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8 | 7 | eqeq1d 2121 |
. 2
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9 | df-tr 3985 |
. . 3
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10 | ssequn1 3210 |
. . 3
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11 | 9, 10 | bitri 183 |
. 2
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12 | 8, 11 | syl6rbbr 198 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-sn 3497 df-pr 3498 df-uni 3701 df-tr 3985 df-suc 4251 |
This theorem is referenced by: onsucuni2 4437 nlimsucg 4439 ctmlemr 6943 nnsf 12880 peano4nninf 12881 |
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