![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elsuci | GIF version |
Description: Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. (Contributed by NM, 6-Jun-1994.) |
Ref | Expression |
---|---|
elsuci | ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4402 | . . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
2 | 1 | eleq2i 2260 | . . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵})) |
3 | elun 3300 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | bitri 184 | . 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) |
5 | elsni 3636 | . . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
6 | 5 | orim2i 762 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
7 | 4, 6 | sylbi 121 | 1 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1364 ∈ wcel 2164 ∪ cun 3151 {csn 3618 suc csuc 4396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-un 3157 df-sn 3624 df-suc 4402 |
This theorem is referenced by: trsucss 4454 onsucelsucexmid 4562 ordsoexmid 4594 ordsuc 4595 ordpwsucexmid 4602 nnsucelsuc 6544 nntri3or 6546 nnmordi 6569 nnaordex 6581 phplem3 6910 nninfninc 7182 nnnninf2 7186 3nelsucpw1 7294 3nsssucpw1 7296 |
Copyright terms: Public domain | W3C validator |