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Mirrors > Home > NFE Home > Th. List > reiotacl2 | GIF version |
Description: Membership law for descriptions. (Contributed by SF, 21-Aug-2011.) |
Ref | Expression |
---|---|
reiotacl2 | ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∈ A ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 2621 | . . 3 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ)) | |
2 | iotacl 4362 | . . 3 ⊢ (∃!x(x ∈ A ∧ φ) → (℩x(x ∈ A ∧ φ)) ∈ {x ∣ (x ∈ A ∧ φ)}) | |
3 | 1, 2 | sylbi 187 | . 2 ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∣ (x ∈ A ∧ φ)}) |
4 | df-rab 2623 | . 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
5 | 3, 4 | syl6eleqr 2444 | 1 ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∈ A ∣ φ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 ∃!weu 2204 {cab 2339 ∃!wreu 2616 {crab 2618 ℩cio 4337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-reu 2621 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-uni 3892 df-iota 4339 |
This theorem is referenced by: reiotacl 4364 |
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