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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 4548. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
sucunielr | ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V) | |
2 | sucexb 4514 | . . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylibr 134 | . . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
4 | sucidg 4434 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴) |
6 | elunii 3829 | . 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 2160 Vcvv 2752 ∪ cuni 3824 suc csuc 4383 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-suc 4389 |
This theorem is referenced by: nnsucuniel 6521 |
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