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| Mirrors > Home > MPE Home > Th. List > elsuci | Structured version Visualization version GIF version | ||
| Description: Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 6-Jun-1994.) |
| Ref | Expression |
|---|---|
| elsuci | ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-suc 6390 | . . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
| 2 | 1 | eleq2i 2833 | . . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵})) |
| 3 | elun 4153 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) | |
| 4 | 2, 3 | bitri 275 | . 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) |
| 5 | elsni 4643 | . . 3 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
| 6 | 5 | orim2i 911 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
| 7 | 4, 6 | sylbi 217 | 1 ⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-suc 6390 |
| This theorem is referenced by: suctr 6470 trsucss 6472 ordnbtwn 6477 suc11 6491 tfrlem11 8428 omordi 8604 nnmordi 8669 pssnn 9208 phplem3OLD 9256 r1sdom 9814 cfsuc 10297 axdc3lem2 10491 axdc3lem4 10493 indpi 10947 constrmon 33785 bnj563 34757 bnj964 34957 ontgval 36432 onsucconni 36438 suctrALT 44846 suctrALT2VD 44856 suctrALT2 44857 suctrALTcf 44942 suctrALTcfVD 44943 suctrALT3 44944 |
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