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Mirrors > Home > MPE Home > Th. List > sucexb | Structured version Visualization version GIF version |
Description: A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
Ref | Expression |
---|---|
sucexb | ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unexb 7451 | . 2 ⊢ ((𝐴 ∈ V ∧ {𝐴} ∈ V) ↔ (𝐴 ∪ {𝐴}) ∈ V) | |
2 | snex 5297 | . . 3 ⊢ {𝐴} ∈ V | |
3 | 2 | biantru 533 | . 2 ⊢ (𝐴 ∈ V ↔ (𝐴 ∈ V ∧ {𝐴} ∈ V)) |
4 | df-suc 6165 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | 4 | eleq1i 2880 | . 2 ⊢ (suc 𝐴 ∈ V ↔ (𝐴 ∪ {𝐴}) ∈ V) |
6 | 1, 3, 5 | 3bitr4i 306 | 1 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 {csn 4525 suc csuc 6161 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-uni 4801 df-suc 6165 |
This theorem is referenced by: sucexg 7505 sucelon 7512 ordsucelsuc 7517 oeordi 8196 suc11reg 9066 rankxpsuc 9295 isf32lem2 9765 limsucncmpi 33906 |
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