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Mirrors > Home > ILE Home > Th. List > elsuc | 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 4321 | . 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 set class |
Syntax hints: ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 Vcvv 2681 suc csuc 4282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-suc 4288 |
This theorem is referenced by: sucel 4327 suctr 4338 0elsucexmid 4475 tfrlemisucaccv 6215 tfr1onlemsucaccv 6231 tfrcllemsucaccv 6244 |
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