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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-tr 4101 |
. . 3
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2 | ssequn1 3305 |
. . 3
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3 | 1, 2 | bitri 184 |
. 2
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4 | df-suc 4370 |
. . . . . 6
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5 | 4 | unieqi 3819 |
. . . . 5
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6 | uniun 3828 |
. . . . 5
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7 | 5, 6 | eqtri 2198 |
. . . 4
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8 | unisng 3826 |
. . . . 5
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9 | 8 | uneq2d 3289 |
. . . 4
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10 | 7, 9 | eqtrid 2222 |
. . 3
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11 | 10 | eqeq1d 2186 |
. 2
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12 | 3, 11 | bitr4id 199 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-uni 3810 df-tr 4101 df-suc 4370 |
This theorem is referenced by: onsucuni2 4562 nlimsucg 4564 ctmlemr 7104 nnnninfeq2 7124 nnsf 14614 peano4nninf 14615 |
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