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Mirrors > Home > ILE Home > Th. List > sucunielr | GIF version |
Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4310. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
sucunielr | ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2621 | . . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V) | |
2 | sucexb 4277 | . . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylibr 132 | . . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
4 | sucidg 4207 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴) |
6 | elunii 3632 | . 2 ⊢ ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → 𝐴 ∈ ∪ 𝐵) | |
7 | 5, 6 | mpancom 413 | 1 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 Vcvv 2612 ∪ cuni 3627 suc csuc 4156 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-uni 3628 df-suc 4162 |
This theorem is referenced by: nnsucuniel 6188 |
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