Theorem dfac15 35802

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

Assertion

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

Distinct variable group:   𝑥,𝑓

Proof of Theorem dfac15

Dummy variables 𝑤 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac10c 10218	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑣 ∈ On 𝑣 ≈ 𝑥)
2		onss 7799	. . . . . 6 ⊢ (𝑣 ∈ On → 𝑣 ⊆ On)
3		ensymb 9030	. . . . . . . 8 ⊢ (𝑣 ≈ 𝑥 ↔ 𝑥 ≈ 𝑣)
4		bren 8984	. . . . . . . 8 ⊢ (𝑥 ≈ 𝑣 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑣)
5	3, 4	bitri 278	. . . . . . 7 ⊢ (𝑣 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑣)
6		f1of1 6823	. . . . . . . . . 10 ⊢ (𝑓:𝑥–1-1-onto→𝑣 → 𝑓:𝑥–1-1→𝑣)
7		f1ss 6785	. . . . . . . . . 10 ⊢ ((𝑓:𝑥–1-1→𝑣 ∧ 𝑣 ⊆ On) → 𝑓:𝑥–1-1→On)
8	6, 7	sylan 592	. . . . . . . . 9 ⊢ ((𝑓:𝑥–1-1-onto→𝑣 ∧ 𝑣 ⊆ On) → 𝑓:𝑥–1-1→On)
9	8	expcom 419	. . . . . . . 8 ⊢ (𝑣 ⊆ On → (𝑓:𝑥–1-1-onto→𝑣 → 𝑓:𝑥–1-1→On))
10	9	eximdv 1949	. . . . . . 7 ⊢ (𝑣 ⊆ On → (∃𝑓 𝑓:𝑥–1-1-onto→𝑣 → ∃𝑓 𝑓:𝑥–1-1→On))
11	5, 10	biimtrid 245	. . . . . 6 ⊢ (𝑣 ⊆ On → (𝑣 ≈ 𝑥 → ∃𝑓 𝑓:𝑥–1-1→On))
12	2, 11	syl 18	. . . . 5 ⊢ (𝑣 ∈ On → (𝑣 ≈ 𝑥 → ∃𝑓 𝑓:𝑥–1-1→On))
13	12	rexlimiv 3156	. . . 4 ⊢ (∃𝑣 ∈ On 𝑣 ≈ 𝑥 → ∃𝑓 𝑓:𝑥–1-1→On)
14	13	alimi 1843	. . 3 ⊢ (∀𝑥∃𝑣 ∈ On 𝑣 ≈ 𝑥 → ∀𝑥∃𝑓 𝑓:𝑥–1-1→On)
15	1, 14	sylbi 220	. 2 ⊢ (CHOICE → ∀𝑥∃𝑓 𝑓:𝑥–1-1→On)
16		eqid 2760	. . . . . . 7 ⊢ {⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} = {⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)}
17	16	onf1we 35764	. . . . . 6 ⊢ (𝑓:𝑥–1-1→On → {⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} We 𝑥)
18		weinxp 5735	. . . . . . 7 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} We 𝑥 ↔ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} ∩ (𝑥 × 𝑥)) We 𝑥)
19		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20	19, 19	xpex 7767	. . . . . . . . 9 ⊢ (𝑥 × 𝑥) ∈ V
21	20	inex2 5277	. . . . . . . 8 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} ∩ (𝑥 × 𝑥)) ∈ V
22		weeq1 5637	. . . . . . . 8 ⊢ (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 3560	. . . . . . 7 ⊢ (({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} ∩ (𝑥 × 𝑥)) We 𝑥 → ∃𝑤 𝑤 We 𝑥)
24	18, 23	sylbi 220	. . . . . 6 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} We 𝑥 → ∃𝑤 𝑤 We 𝑥)
25	17, 24	syl 18	. . . . 5 ⊢ (𝑓:𝑥–1-1→On → ∃𝑤 𝑤 We 𝑥)
26	25	exlimiv 1962	. . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1→On → ∃𝑤 𝑤 We 𝑥)
27	26	alimi 1843	. . 3 ⊢ (∀𝑥∃𝑓 𝑓:𝑥–1-1→On → ∀𝑥∃𝑤 𝑤 We 𝑥)
28		dfac8 10215	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑤 𝑤 We 𝑥)
29	27, 28	sylibr 237	. 2 ⊢ (∀𝑥∃𝑓 𝑓:𝑥–1-1→On → CHOICE)
30	15, 29	impbii 212	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓 𝑓:𝑥–1-1→On)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102  {copab 5166   We wwe 5602   × cxp 5648  Oncon0 6362  –1-1→wf1 6535  –1-1-onto→wf1o 6537  ‘cfv 6538   ≈ cen 8971  CHOICEwac 10195

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-er 8718  df-en 8975  df-card 10021  df-ac 10196

This theorem is used by:  onf1oracwevd  35964  onprcf1of1orbd  35966