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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 4263 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | unieqi 3716 | . . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
3 | uniun 3725 | . . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
4 | 2, 3 | eqtri 2138 | . . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴}) |
5 | unisng 3723 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
6 | 5 | uneq2d 3200 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)) |
7 | 4, 6 | syl5eq 2162 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) |
8 | 7 | eqeq1d 2126 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)) |
9 | df-tr 3997 | . . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
10 | ssequn1 3216 | . . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) | |
11 | 9, 10 | bitri 183 | . 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) |
12 | 8, 11 | syl6rbbr 198 | 1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∪ cun 3039 ⊆ wss 3041 {csn 3497 ∪ cuni 3706 Tr wtr 3996 suc csuc 4257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-suc 4263 |
This theorem is referenced by: onsucuni2 4449 nlimsucg 4451 ctmlemr 6961 nnsf 13126 peano4nninf 13127 |
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