Theorem ween

Description: A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013)

	Ref	Expression
Assertion	ween	⊢ ( 𝐴 ∈ dom card ↔ ∃ 𝑟 𝑟 We 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dfac8b	⊢ ( 𝐴 ∈ dom card → ∃ 𝑟 𝑟 We 𝐴 )
2		weso	⊢ ( 𝑟 We 𝐴 → 𝑟 Or 𝐴 )
3		vex	⊢ 𝑟 ∈ V
4		soex	⊢ ( ( 𝑟 Or 𝐴 ∧ 𝑟 ∈ V ) → 𝐴 ∈ V )
5	2 3 4	sylancl	⊢ ( 𝑟 We 𝐴 → 𝐴 ∈ V )
6	5	exlimiv	⊢ ( ∃ 𝑟 𝑟 We 𝐴 → 𝐴 ∈ V )
7		unipw	⊢ ∪ 𝒫 𝐴 = 𝐴
8		weeq2	⊢ ( ∪ 𝒫 𝐴 = 𝐴 → ( 𝑟 We ∪ 𝒫 𝐴 ↔ 𝑟 We 𝐴 ) )
9	7 8	ax-mp	⊢ ( 𝑟 We ∪ 𝒫 𝐴 ↔ 𝑟 We 𝐴 )
10	9	exbii	⊢ ( ∃ 𝑟 𝑟 We ∪ 𝒫 𝐴 ↔ ∃ 𝑟 𝑟 We 𝐴 )
11	10	biimpri	⊢ ( ∃ 𝑟 𝑟 We 𝐴 → ∃ 𝑟 𝑟 We ∪ 𝒫 𝐴 )
12		pwexg	⊢ ( 𝐴 ∈ V → 𝒫 𝐴 ∈ V )
13		dfac8c	⊢ ( 𝒫 𝐴 ∈ V → ( ∃ 𝑟 𝑟 We ∪ 𝒫 𝐴 → ∃ 𝑓 ∀ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) ) )
14	12 13	syl	⊢ ( 𝐴 ∈ V → ( ∃ 𝑟 𝑟 We ∪ 𝒫 𝐴 → ∃ 𝑓 ∀ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) ) )
15		dfac8a	⊢ ( 𝐴 ∈ V → ( ∃ 𝑓 ∀ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≠ ∅ → ( 𝑓 ‘ 𝑥 ) ∈ 𝑥 ) → 𝐴 ∈ dom card ) )
16	14 15	syld	⊢ ( 𝐴 ∈ V → ( ∃ 𝑟 𝑟 We ∪ 𝒫 𝐴 → 𝐴 ∈ dom card ) )
17	6 11 16	sylc	⊢ ( ∃ 𝑟 𝑟 We 𝐴 → 𝐴 ∈ dom card )
18	1 17	impbii	⊢ ( 𝐴 ∈ dom card ↔ ∃ 𝑟 𝑟 We 𝐴 )