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Mirrors > Home > ILE Home > Th. List > riotaprop | GIF version |
Description: Properties of a restricted definite description operator. Todo (df-riota 5411 update): can some uses of riota2f 5432 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
Ref | Expression |
---|---|
riotaprop.0 | ⊢ Ⅎxψ |
riotaprop.1 | ⊢ B = (℩x ∈ A φ) |
riotaprop.2 | ⊢ (x = B → (φ ↔ ψ)) |
Ref | Expression |
---|---|
riotaprop | ⊢ (∃!x ∈ A φ → (B ∈ A ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaprop.1 | . . 3 ⊢ B = (℩x ∈ A φ) | |
2 | riotacl 5425 | . . 3 ⊢ (∃!x ∈ A φ → (℩x ∈ A φ) ∈ A) | |
3 | 1, 2 | syl5eqel 2121 | . 2 ⊢ (∃!x ∈ A φ → B ∈ A) |
4 | 1 | eqcomi 2041 | . . . 4 ⊢ (℩x ∈ A φ) = B |
5 | nfriota1 5418 | . . . . . 6 ⊢ Ⅎx(℩x ∈ A φ) | |
6 | 1, 5 | nfcxfr 2172 | . . . . 5 ⊢ ℲxB |
7 | riotaprop.0 | . . . . 5 ⊢ Ⅎxψ | |
8 | riotaprop.2 | . . . . 5 ⊢ (x = B → (φ ↔ ψ)) | |
9 | 6, 7, 8 | riota2f 5432 | . . . 4 ⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → (ψ ↔ (℩x ∈ A φ) = B)) |
10 | 4, 9 | mpbiri 157 | . . 3 ⊢ ((B ∈ A ∧ ∃!x ∈ A φ) → ψ) |
11 | 3, 10 | mpancom 399 | . 2 ⊢ (∃!x ∈ A φ → ψ) |
12 | 3, 11 | jca 290 | 1 ⊢ (∃!x ∈ A φ → (B ∈ A ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 Ⅎwnf 1346 ∈ wcel 1390 ∃!wreu 2302 ℩crio 5410 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-uni 3572 df-iota 4810 df-riota 5411 |
This theorem is referenced by: (None) |
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