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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 4196 |
. . . . . 6
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2 | 1 | unieqi 3661 |
. . . . 5
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3 | uniun 3670 |
. . . . 5
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4 | 2, 3 | eqtri 2108 |
. . . 4
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5 | unisng 3668 |
. . . . 5
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6 | 5 | uneq2d 3154 |
. . . 4
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7 | 4, 6 | syl5eq 2132 |
. . 3
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8 | 7 | eqeq1d 2096 |
. 2
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9 | df-tr 3935 |
. . 3
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10 | ssequn1 3170 |
. . 3
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11 | 9, 10 | bitri 182 |
. 2
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12 | 8, 11 | syl6rbbr 197 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-sn 3450 df-pr 3451 df-uni 3652 df-tr 3935 df-suc 4196 |
This theorem is referenced by: onsucuni2 4378 nlimsucg 4380 nnsf 11778 peano4nninf 11779 |
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