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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 4389 | . 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 703 = wceq 1348 ∈ wcel 2141 Vcvv 2730 suc csuc 4350 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-suc 4356 |
This theorem is referenced by: sucel 4395 suctr 4406 0elsucexmid 4549 tfrlemisucaccv 6304 tfr1onlemsucaccv 6320 tfrcllemsucaccv 6333 |
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