Theorem ac10ct 10143

Description: A proof of the well-ordering theorem weth 10603, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.)

Assertion

Ref	Expression
ac10ct	⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem ac10ct

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

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	brdom 9018	. . . . 5 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝑦)
3		onss 7826	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶1-1𝑦 → (𝑦 ∈ On → 𝑦 ⊆ On))
5		epweon 7816	. . . . . . . . . 10 ⊢ E We On
6		wess 5642	. . . . . . . . . 10 ⊢ (𝑦 ⊆ On → ( E We On → E We 𝑦))
7	4, 5, 6	syl6mpi 68	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶1-1𝑦 → (𝑦 ∈ On → E We 𝑦))
8	7	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴⟶1-1𝑦) → (𝑦 ∈ On → E We 𝑦))
9		eqid 2763	. . . . . . . . . 10 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}
10	9	f1we 7389	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶1-1𝑦 → ( E We 𝑦 → {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴))
11		weinxp 5742	. . . . . . . . . 10 ⊢ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
12		reldom 9010	. . . . . . . . . . . 12 ⊢ Rel ≼
13	12	brrelex1i 5712	. . . . . . . . . . 11 ⊢ (𝐴 ≼ 𝑦 → 𝐴 ∈ V)
14		sqxpexg 7796	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
15		inex2g 5285	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
16		weeq1 5643	. . . . . . . . . . . 12 ⊢ (𝑥 = ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴))
17	16	spcegv 3555	. . . . . . . . . . 11 ⊢ (({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) ∈ V → (({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
18	13, 14, 15, 17	4syl 20	. . . . . . . . . 10 ⊢ (𝐴 ≼ 𝑦 → (({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
19	11, 18	biimtrid 245	. . . . . . . . 9 ⊢ (𝐴 ≼ 𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
20	10, 19	sylan9r 518	. . . . . . . 8 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴⟶1-1𝑦) → ( E We 𝑦 → ∃𝑥 𝑥 We 𝐴))
21	8, 20	syld 48	. . . . . . 7 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴⟶1-1𝑦) → (𝑦 ∈ On → ∃𝑥 𝑥 We 𝐴))
22	21	impancom 457	. . . . . 6 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (𝑓:𝐴⟶1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
23	22	exlimdv 1968	. . . . 5 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (∃𝑓 𝑓:𝐴⟶1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
24	2, 23	biimtrid 245	. . . 4 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴))
25	24	ex 418	. . 3 ⊢ (𝐴 ≼ 𝑦 → (𝑦 ∈ On → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)))
26	25	pm2.43b 56	. 2 ⊢ (𝑦 ∈ On → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴))
27	26	rexlimiv 3160	1 ⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5106  {copab 5171   E cep 5555   We wwe 5608   × cxp 5654  Oncon0 6383  ⟶1-1wf1 6558  ‘cfv 6561   ≼ cdom 9002

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  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-br 5107  df-opab 5172  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-dom 9006

This theorem is used by:  ondomen  10146  dfac16  35859