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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 4074 | . . . 4 ⊢ suc B = (B ∪ {B}) | |
2 | 1 | eleq2i 2101 | . . 3 ⊢ (A ∈ suc B ↔ A ∈ (B ∪ {B})) |
3 | elun 3078 | . . 3 ⊢ (A ∈ (B ∪ {B}) ↔ (A ∈ B ∨ A ∈ {B})) | |
4 | 2, 3 | bitri 173 | . 2 ⊢ (A ∈ suc B ↔ (A ∈ B ∨ A ∈ {B})) |
5 | elsni 3391 | . . 3 ⊢ (A ∈ {B} → A = B) | |
6 | 5 | orim2i 677 | . 2 ⊢ ((A ∈ B ∨ A ∈ {B}) → (A ∈ B ∨ A = B)) |
7 | 4, 6 | sylbi 114 | 1 ⊢ (A ∈ suc B → (A ∈ B ∨ A = B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ∪ cun 2909 {csn 3367 suc csuc 4068 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-suc 4074 |
This theorem is referenced by: trsucss 4126 onsucelsucexmid 4215 ordsoexmid 4240 ordsuc 4241 ordpwsucexmid 4246 nnsucelsuc 6009 nntri3or 6011 nnmordi 6025 nnaordex 6036 |
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