Theorem riotaclb 6548

Description: Bidirectional closure of restricted iota when domain is not empty. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) (Revised by NM, 13-Sep-2018.)

Assertion

Ref	Expression
riotaclb	⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riotaclb

Step	Hyp	Ref	Expression
1		riotacl 6525	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
2		riotaund 6546	. . . . . 6 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∅)
3	2	eleq1d 2672	. . . . 5 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ∅ ∈ 𝐴))
4	3	notbid 307	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐴 𝜑 → (¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴))
5	4	biimprcd 239	. . 3 ⊢ (¬ ∅ ∈ 𝐴 → (¬ ∃!𝑥 ∈ 𝐴 𝜑 → ¬ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
6	5	con4d 113	. 2 ⊢ (¬ ∅ ∈ 𝐴 → ((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴 → ∃!𝑥 ∈ 𝐴 𝜑))
7	1, 6	impbid2 215	1 ⊢ (¬ ∅ ∈ 𝐴 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∈ wcel 1977  ∃!wreu 2898  ∅c0 3874  ℩crio 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-riota 6511

This theorem is referenced by: (None)