![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4161 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | unieqi 3637 | . . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
3 | uniun 3646 | . . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
4 | 2, 3 | eqtri 2103 | . . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴}) |
5 | unisng 3644 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
6 | 5 | uneq2d 3138 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)) |
7 | 4, 6 | syl5eq 2127 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) |
8 | 7 | eqeq1d 2091 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)) |
9 | df-tr 3902 | . . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
10 | ssequn1 3154 | . . 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 1285 ∈ wcel 1434 ∪ cun 2982 ⊆ wss 2984 {csn 3422 ∪ cuni 3627 Tr wtr 3901 suc csuc 4155 |
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 4161 |
This theorem is referenced by: onsucuni2 4342 nlimsucg 4344 |
Copyright terms: Public domain | W3C validator |