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Mirrors > Home > ILE Home > Th. List > sucel | GIF version |
Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.) |
Ref | Expression |
---|---|
sucel | ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 2461 | . 2 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴) | |
2 | dfcleq 2131 | . . . 4 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴)) | |
3 | vex 2684 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | elsuc 4323 | . . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) |
5 | 4 | bibi2i 226 | . . . . 5 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) |
6 | 5 | albii 1446 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) |
7 | 2, 6 | bitri 183 | . . 3 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) |
8 | 7 | rexbii 2440 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) |
9 | 1, 8 | bitri 183 | 1 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 697 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 suc csuc 4282 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-sn 3528 df-suc 4288 |
This theorem is referenced by: (None) |
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