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Mirrors > Home > MPE Home > Th. List > sucexg | Structured version Visualization version GIF version |
Description: The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
sucexg | ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | sucexb 6901 | . 2 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylib 207 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
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-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 df-suc 5646 |
This theorem is referenced by: sucex 6903 suceloni 6905 hsmexlem1 9131 dfon2lem3 30934 |
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