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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 4526. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
sucunielr | ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 | . . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V) | |
2 | sucexb 4492 | . . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylibr 134 | . . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
4 | sucidg 4412 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴) |
6 | elunii 3812 | . 2 ⊢ ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → 𝐴 ∈ ∪ 𝐵) | |
7 | 5, 6 | mpancom 422 | 1 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ∪ cuni 3807 suc csuc 4361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-uni 3808 df-suc 4367 |
This theorem is referenced by: nnsucuniel 6489 |
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