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Mirrors > Home > ILE Home > Th. List > elsucg | GIF version |
Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.) |
Ref | Expression |
---|---|
elsucg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4126 | . . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
2 | 1 | eleq2i 2145 | . . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵})) |
3 | elun 3113 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | bitri 182 | . 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) |
5 | elsng 3413 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
6 | 5 | orbi2d 736 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
7 | 4, 6 | syl5bb 190 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 ∪ cun 2971 {csn 3398 suc csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-suc 4126 |
This theorem is referenced by: elsuc 4161 elelsuc 4164 sucidg 4171 onsucelsucr 4252 onsucsssucexmid 4270 suc11g 4300 nlt1pig 6531 bj-peano4 10750 |
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