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Mirrors > Home > ILE Home > Th. List > unisucg | 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 Jim Kingdon, 18-Aug-2019.) |
Ref | Expression |
---|---|
unisucg | ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4126 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | unieqi 3611 | . . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
3 | uniun 3620 | . . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
4 | 2, 3 | eqtri 2101 | . . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴}) |
5 | unisng 3618 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
6 | 5 | uneq2d 3126 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)) |
7 | 4, 6 | syl5eq 2125 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) |
8 | 7 | eqeq1d 2089 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)) |
9 | df-tr 3876 | . . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
10 | ssequn1 3142 | . . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) | |
11 | 9, 10 | bitri 182 | . 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) |
12 | 8, 11 | syl6rbbr 197 | 1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∪ cun 2971 ⊆ wss 2973 {csn 3398 ∪ cuni 3601 Tr wtr 3875 suc csuc 4120 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-suc 4126 |
This theorem is referenced by: onsucuni2 4307 nlimsucg 4309 |
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