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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, inference form. 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 | unisuc.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | unisucg 6393 | . 2 ⊢ (𝐴 ∈ V → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cuni 4863 Tr wtr 5220 suc csuc 6317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3445 df-un 3913 df-in 3915 df-ss 3925 df-sn 4585 df-pr 4587 df-uni 4864 df-tr 5221 df-suc 6321 |
This theorem is referenced by: ordunisuc 7759 |
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