Theorem dfac15

Description: Axiom of Choice equivalent: every set can be mapped one-to-one into the ordinals. (Contributed by BTernaryTau, 16-Sep-2026)

	Ref	Expression
Assertion	dfac15	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑓 𝑓 : 𝑥 –1-1→ On )

Proof

Step	Hyp	Ref	Expression
1		dfac10c	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑣 ∈ On 𝑣 ≈ 𝑥 )
2		onss	⊢ ( 𝑣 ∈ On → 𝑣 ⊆ On )
3		ensymb	⊢ ( 𝑣 ≈ 𝑥 ↔ 𝑥 ≈ 𝑣 )
4		bren	⊢ ( 𝑥 ≈ 𝑣 ↔ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑣 )
5	3 4	bitri	⊢ ( 𝑣 ≈ 𝑥 ↔ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑣 )
6		f1of1	⊢ ( 𝑓 : 𝑥 –1-1-onto→ 𝑣 → 𝑓 : 𝑥 –1-1→ 𝑣 )
7		f1ss	⊢ ( ( 𝑓 : 𝑥 –1-1→ 𝑣 ∧ 𝑣 ⊆ On ) → 𝑓 : 𝑥 –1-1→ On )
8	6 7	sylan	⊢ ( ( 𝑓 : 𝑥 –1-1-onto→ 𝑣 ∧ 𝑣 ⊆ On ) → 𝑓 : 𝑥 –1-1→ On )
9	8	expcom	⊢ ( 𝑣 ⊆ On → ( 𝑓 : 𝑥 –1-1-onto→ 𝑣 → 𝑓 : 𝑥 –1-1→ On ) )
10	9	eximdv	⊢ ( 𝑣 ⊆ On → ( ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑣 → ∃ 𝑓 𝑓 : 𝑥 –1-1→ On ) )
11	5 10	biimtrid	⊢ ( 𝑣 ⊆ On → ( 𝑣 ≈ 𝑥 → ∃ 𝑓 𝑓 : 𝑥 –1-1→ On ) )
12	2 11	syl	⊢ ( 𝑣 ∈ On → ( 𝑣 ≈ 𝑥 → ∃ 𝑓 𝑓 : 𝑥 –1-1→ On ) )
13	12	rexlimiv	⊢ ( ∃ 𝑣 ∈ On 𝑣 ≈ 𝑥 → ∃ 𝑓 𝑓 : 𝑥 –1-1→ On )
14	13	alimi	⊢ ( ∀ 𝑥 ∃ 𝑣 ∈ On 𝑣 ≈ 𝑥 → ∀ 𝑥 ∃ 𝑓 𝑓 : 𝑥 –1-1→ On )
15	1 14	sylbi	⊢ ( CHOICE → ∀ 𝑥 ∃ 𝑓 𝑓 : 𝑥 –1-1→ On )
16		eqid	⊢ { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) }
17	16	onf1we	⊢ ( 𝑓 : 𝑥 –1-1→ On → { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } We 𝑥 )
18		weinxp	⊢ ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } We 𝑥 ↔ ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } ∩ ( 𝑥 × 𝑥 ) ) We 𝑥 )
19		vex	⊢ 𝑥 ∈ V
20	19 19	xpex	⊢ ( 𝑥 × 𝑥 ) ∈ V
21	20	inex2	⊢ ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } ∩ ( 𝑥 × 𝑥 ) ) ∈ V
22		weeq1	⊢ ( 𝑤 = ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } ∩ ( 𝑥 × 𝑥 ) ) → ( 𝑤 We 𝑥 ↔ ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } ∩ ( 𝑥 × 𝑥 ) ) We 𝑥 ) )
23	21 22	spcev	⊢ ( ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } ∩ ( 𝑥 × 𝑥 ) ) We 𝑥 → ∃ 𝑤 𝑤 We 𝑥 )
24	18 23	sylbi	⊢ ( { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑓 ‘ 𝑧 ) } We 𝑥 → ∃ 𝑤 𝑤 We 𝑥 )
25	17 24	syl	⊢ ( 𝑓 : 𝑥 –1-1→ On → ∃ 𝑤 𝑤 We 𝑥 )
26	25	exlimiv	⊢ ( ∃ 𝑓 𝑓 : 𝑥 –1-1→ On → ∃ 𝑤 𝑤 We 𝑥 )
27	26	alimi	⊢ ( ∀ 𝑥 ∃ 𝑓 𝑓 : 𝑥 –1-1→ On → ∀ 𝑥 ∃ 𝑤 𝑤 We 𝑥 )
28		dfac8	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑤 𝑤 We 𝑥 )
29	27 28	sylibr	⊢ ( ∀ 𝑥 ∃ 𝑓 𝑓 : 𝑥 –1-1→ On → CHOICE )
30	15 29	impbii	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑓 𝑓 : 𝑥 –1-1→ On )