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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-tr 4099 | . . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
2 | ssequn1 3305 | . . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) | |
3 | 1, 2 | bitri 184 | . 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) |
4 | df-suc 4367 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | unieqi 3817 | . . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
6 | uniun 3826 | . . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
7 | 5, 6 | eqtri 2198 | . . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴}) |
8 | unisng 3824 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | |
9 | 8 | uneq2d 3289 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)) |
10 | 7, 9 | eqtrid 2222 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)) |
11 | 10 | eqeq1d 2186 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)) |
12 | 3, 11 | bitr4id 199 | 1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 ⊆ wss 3129 {csn 3591 ∪ cuni 3807 Tr wtr 4098 suc csuc 4361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-uni 3808 df-tr 4099 df-suc 4367 |
This theorem is referenced by: onsucuni2 4559 nlimsucg 4561 ctmlemr 7100 nnnninfeq2 7120 nnsf 14377 peano4nninf 14378 |
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