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Mirrors > Home > MPE Home > Th. List > unisuc | Structured version Visualization version 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 3745 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) | |
2 | df-tr 4681 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | df-suc 5646 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | unieqi 4381 | . . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
5 | uniun 4392 | . . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
6 | unisuc.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
7 | 6 | unisn 4387 | . . . . 5 ⊢ ∪ {𝐴} = 𝐴 |
8 | 7 | uneq2i 3726 | . . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴) |
9 | 4, 5, 8 | 3eqtri 2636 | . . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴) |
10 | 9 | eqeq1i 2615 | . 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) |
11 | 1, 2, 10 | 3bitr4i 291 | 1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 {csn 4125 ∪ cuni 4372 Tr wtr 4680 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-uni 4373 df-tr 4681 df-suc 5646 |
This theorem is referenced by: onunisuci 5758 ordunisuc 6924 |
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