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Mirrors > Home > MPE Home > Th. List > elsuc | Structured version Visualization version GIF version |
Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.) |
Ref | Expression |
---|---|
elsuc.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elsuc | ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuc.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elsucg 6226 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-suc 6165 |
This theorem is referenced by: sucel 6232 limsssuc 7545 omsmolem 8263 cantnfle 9118 infxpenlem 9424 inatsk 10189 untsucf 33049 dfon2lem7 33147 nolesgn2ores 33292 rdgssun 34795 |
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