Theorem dfac15 35845

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⟶1-1On)

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 10244	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑣 ∈ On 𝑣 ≈ 𝑥)
2		onss 7823	. . . . . 6 ⊢ (𝑣 ∈ On → 𝑣 ⊆ On)
3		ensymb 9057	. . . . . . . 8 ⊢ (𝑣 ≈ 𝑥 ↔ 𝑥 ≈ 𝑣)
4		bren 9011	. . . . . . . 8 ⊢ (𝑥 ≈ 𝑣 ↔ ∃𝑓 𝑓:𝑥⟶1-1onto𝑣)
5	3, 4	bitri 278	. . . . . . 7 ⊢ (𝑣 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝑥⟶1-1onto𝑣)
6		f1of1 6845	. . . . . . . . . 10 ⊢ (𝑓:𝑥⟶1-1onto𝑣 → 𝑓:𝑥⟶1-1𝑣)
7		f1ss 6807	. . . . . . . . . 10 ⊢ ((𝑓:𝑥⟶1-1𝑣 ∧ 𝑣 ⊆ On) → 𝑓:𝑥⟶1-1On)
8	6, 7	sylan 592	. . . . . . . . 9 ⊢ ((𝑓:𝑥⟶1-1onto𝑣 ∧ 𝑣 ⊆ On) → 𝑓:𝑥⟶1-1On)
9	8	expcom 419	. . . . . . . 8 ⊢ (𝑣 ⊆ On → (𝑓:𝑥⟶1-1onto𝑣 → 𝑓:𝑥⟶1-1On))
10	9	eximdv 1952	. . . . . . 7 ⊢ (𝑣 ⊆ On → (∃𝑓 𝑓:𝑥⟶1-1onto𝑣 → ∃𝑓 𝑓:𝑥⟶1-1On))
11	5, 10	biimtrid 245	. . . . . 6 ⊢ (𝑣 ⊆ On → (𝑣 ≈ 𝑥 → ∃𝑓 𝑓:𝑥⟶1-1On))
12	2, 11	syl 18	. . . . 5 ⊢ (𝑣 ∈ On → (𝑣 ≈ 𝑥 → ∃𝑓 𝑓:𝑥⟶1-1On))
13	12	rexlimiv 3160	. . . 4 ⊢ (∃𝑣 ∈ On 𝑣 ≈ 𝑥 → ∃𝑓 𝑓:𝑥⟶1-1On)
14	13	alimi 1844	. . 3 ⊢ (∀𝑥∃𝑣 ∈ On 𝑣 ≈ 𝑥 → ∀𝑥∃𝑓 𝑓:𝑥⟶1-1On)
15	1, 14	sylbi 220	. 2 ⊢ (CHOICE → ∀𝑥∃𝑓 𝑓:𝑥⟶1-1On)
16		eqid 2763	. . . . . . 7 ⊢ {⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} = {⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)}
17	16	onf1we 35803	. . . . . 6 ⊢ (𝑓:𝑥⟶1-1On → {⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} We 𝑥)
18		weinxp 5741	. . . . . . 7 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} We 𝑥 ↔ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} ∩ (𝑥 × 𝑥)) We 𝑥)
19		vex 3458	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20	19, 19	xpex 7791	. . . . . . . . 9 ⊢ (𝑥 × 𝑥) ∈ V
21	20	inex2 5282	. . . . . . . 8 ⊢ ({⟨𝑦, 𝑧⟩ ∣ (𝑓‘𝑦) ∈ (𝑓‘𝑧)} ∩ (𝑥 × 𝑥)) ∈ V
22		weeq1 5642	. . . . . . . 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 3564	. . . . . . 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⟶1-1On → ∃𝑤 𝑤 We 𝑥)
26	25	exlimiv 1965	. . . 4 ⊢ (∃𝑓 𝑓:𝑥⟶1-1On → ∃𝑤 𝑤 We 𝑥)
27	26	alimi 1844	. . 3 ⊢ (∀𝑥∃𝑓 𝑓:𝑥⟶1-1On → ∀𝑥∃𝑤 𝑤 We 𝑥)
28		dfac8 10241	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑤 𝑤 We 𝑥)
29	27, 28	sylibr 237	. 2 ⊢ (∀𝑥∃𝑓 𝑓:𝑥⟶1-1On → CHOICE)
30	15, 29	impbii 212	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓 𝑓:𝑥⟶1-1On)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5106  {copab 5170   We wwe 5607   × cxp 5653  Oncon0 6380  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557  ‘cfv 6558   ≈ cen 8998  CHOICEwac 10221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-er 8745  df-en 9002  df-card 10047  df-ac 10222

This theorem is used by:  onf1oracwevd  36052  onprcf1of1orbd  36054