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Mirrors > Home > ILE Home > Th. List > unisuc | 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 NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
unisuc.1 |
Ref | Expression |
---|---|
unisuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 3113 | . 2 | |
2 | df-tr 3855 | . 2 | |
3 | df-suc 4108 | . . . . 5 | |
4 | 3 | unieqi 3590 | . . . 4 |
5 | uniun 3599 | . . . 4 | |
6 | unisuc.1 | . . . . . 6 | |
7 | 6 | unisn 3596 | . . . . 5 |
8 | 7 | uneq2i 3094 | . . . 4 |
9 | 4, 5, 8 | 3eqtri 2064 | . . 3 |
10 | 9 | eqeq1i 2047 | . 2 |
11 | 1, 2, 10 | 3bitr4i 201 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 98 wceq 1243 wcel 1393 cvv 2557 cun 2915 wss 2917 csn 3375 cuni 3580 wtr 3854 csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-suc 4108 |
This theorem is referenced by: onunisuci 4169 ordsucunielexmid 4256 tfrexlem 5948 |
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