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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 4301 | . . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
2 | 1 | eleq2i 2207 | . . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵})) |
3 | elun 3222 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | bitri 183 | . 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) |
5 | elsni 3550 | . . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
6 | 5 | orim2i 751 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
7 | 4, 6 | sylbi 120 | 1 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1332 ∈ wcel 1481 ∪ cun 3074 {csn 3532 suc csuc 4295 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-suc 4301 |
This theorem is referenced by: trsucss 4353 onsucelsucexmid 4453 ordsoexmid 4485 ordsuc 4486 ordpwsucexmid 4493 nnsucelsuc 6395 nntri3or 6397 nnmordi 6420 nnaordex 6431 phplem3 6756 |
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