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Mirrors > Home > MPE Home > Th. List > elsucg | Structured version Visualization version 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 5646 | . . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
2 | 1 | eleq2i 2680 | . . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵})) |
3 | elun 3715 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | bitri 263 | . 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵})) |
5 | elsng 4139 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
6 | 5 | orbi2d 734 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
7 | 4, 6 | syl5bb 271 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {csn 4125 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-suc 5646 |
This theorem is referenced by: elsuc 5711 elelsuc 5714 sucidg 5720 ordsssuc 5729 ordsucelsuc 6914 suc11reg 8399 nlt1pi 9607 |
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