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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 4158 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) | |
2 | df-tr 5175 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | df-suc 6199 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | unieqi 4853 | . . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}) |
5 | uniun 4863 | . . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}) | |
6 | unisuc.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
7 | 6 | unisn 4860 | . . . . 5 ⊢ ∪ {𝐴} = 𝐴 |
8 | 7 | uneq2i 4138 | . . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴) |
9 | 4, 5, 8 | 3eqtri 2850 | . . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴) |
10 | 9 | eqeq1i 2828 | . 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴) |
11 | 1, 2, 10 | 3bitr4i 305 | 1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 {csn 4569 ∪ cuni 4840 Tr wtr 5174 suc csuc 6195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-sn 4570 df-pr 4572 df-uni 4841 df-tr 5175 df-suc 6199 |
This theorem is referenced by: onunisuci 6306 ordunisuc 7549 |
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