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Mirrors > Home > ILE Home > Th. List > unisuc | GIF 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 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
unisuc | ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 3154 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) | |
2 | df-tr 3902 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | df-suc 4162 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | unieqi 3637 | . . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
5 | uniun 3646 | . . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
6 | unisuc.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
7 | 6 | unisn 3643 | . . . . 5 ⊢ ∪ {𝐴} = 𝐴 |
8 | 7 | uneq2i 3135 | . . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴) |
9 | 4, 5, 8 | 3eqtri 2107 | . . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴) |
10 | 9 | eqeq1i 2090 | . 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) |
11 | 1, 2, 10 | 3bitr4i 210 | 1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1285 ∈ wcel 1434 Vcvv 2612 ∪ cun 2982 ⊆ wss 2984 {csn 3422 ∪ cuni 3627 Tr wtr 3901 suc csuc 4156 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-sn 3428 df-pr 3429 df-uni 3628 df-tr 3902 df-suc 4162 |
This theorem is referenced by: onunisuci 4223 ordsucunielexmid 4310 tfrexlem 6031 nnsucuniel 6188 |
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