![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4406. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
sucunielr | ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2668 | . . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V) | |
2 | sucexb 4373 | . . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | sylibr 133 | . . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
4 | sucidg 4298 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴) |
6 | elunii 3707 | . 2 ⊢ ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → 𝐴 ∈ ∪ 𝐵) | |
7 | 5, 6 | mpancom 416 | 1 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 Vcvv 2657 ∪ cuni 3702 suc csuc 4247 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-uni 3703 df-suc 4253 |
This theorem is referenced by: nnsucuniel 6345 |
Copyright terms: Public domain | W3C validator |