Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > tpid3g | GIF version |
Description: Closed theorem form of tpid3 3639. (Contributed by Alan Sare, 24-Oct-2011.) |
Ref | Expression |
---|---|
tpid3g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2700 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
2 | 3mix3 1152 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)) | |
3 | 2 | a1i 9 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))) |
4 | abid 2127 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} ↔ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)) | |
5 | 3, 4 | syl6ibr 161 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)})) |
6 | dftp2 3572 | . . . . . 6 ⊢ {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} | |
7 | 6 | eleq2i 2206 | . . . . 5 ⊢ (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}) |
8 | 5, 7 | syl6ibr 161 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ {𝐶, 𝐷, 𝐴})) |
9 | eleq1 2202 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴})) | |
10 | 8, 9 | mpbidi 150 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})) |
11 | 10 | exlimdv 1791 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})) |
12 | 1, 11 | mpd 13 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 961 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2125 {ctp 3529 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-tp 3535 |
This theorem is referenced by: rngmulrg 12077 srngmulrd 12084 lmodscad 12095 ipsmulrd 12103 ipsipd 12106 topgrptsetd 12113 |
Copyright terms: Public domain | W3C validator |