Theorem ac10ct 10103

Description: A proof of the well-ordering theorem weth 10564, 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 3492	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	brdom 9020	. . . . 5 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)
3		f1f 6817	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴⟶𝑦)
4	3	frnd 6755	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1→𝑦 → ran 𝑓 ⊆ 𝑦)
5		onss 7820	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
6		sstr2 4015	. . . . . . . . . . 11 ⊢ (ran 𝑓 ⊆ 𝑦 → (𝑦 ⊆ On → ran 𝑓 ⊆ On))
7	4, 5, 6	syl2im 40	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝑦 → (𝑦 ∈ On → ran 𝑓 ⊆ On))
8		epweon 7810	. . . . . . . . . 10 ⊢ E We On
9		wess 5686	. . . . . . . . . 10 ⊢ (ran 𝑓 ⊆ On → ( E We On → E We ran 𝑓))
10	7, 8, 9	syl6mpi 67	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝑦 → (𝑦 ∈ On → E We ran 𝑓))
11	10	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) → (𝑦 ∈ On → E We ran 𝑓))
12		f1f1orn 6873	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴–1-1-onto→ran 𝑓)
13		eqid 2740	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}
14	13	f1owe 7389	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → ( E We ran 𝑓 → {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴))
15	12, 14	syl 17	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1→𝑦 → ( E We ran 𝑓 → {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴))
16		weinxp 5784	. . . . . . . . . 10 ⊢ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
17		reldom 9009	. . . . . . . . . . . 12 ⊢ Rel ≼
18	17	brrelex1i 5756	. . . . . . . . . . 11 ⊢ (𝐴 ≼ 𝑦 → 𝐴 ∈ V)
19		sqxpexg 7790	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
20		incom 4230	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴))
21		inex1g 5337	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) ∈ V)
22	20, 21	eqeltrrid 2849	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
23		weeq1 5687	. . . . . . . . . . . 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))
24	23	spcegv 3610	. . . . . . . . . . 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))
25	18, 19, 22, 24	4syl 19	. . . . . . . . . 10 ⊢ (𝐴 ≼ 𝑦 → (({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
26	16, 25	biimtrid 242	. . . . . . . . 9 ⊢ (𝐴 ≼ 𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
27	15, 26	sylan9r 508	. . . . . . . 8 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) → ( E We ran 𝑓 → ∃𝑥 𝑥 We 𝐴))
28	11, 27	syld 47	. . . . . . 7 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) → (𝑦 ∈ On → ∃𝑥 𝑥 We 𝐴))
29	28	impancom 451	. . . . . 6 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (𝑓:𝐴–1-1→𝑦 → ∃𝑥 𝑥 We 𝐴))
30	29	exlimdv 1932	. . . . 5 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (∃𝑓 𝑓:𝐴–1-1→𝑦 → ∃𝑥 𝑥 We 𝐴))
31	2, 30	biimtrid 242	. . . 4 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴))
32	31	ex 412	. . 3 ⊢ (𝐴 ≼ 𝑦 → (𝑦 ∈ On → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)))
33	32	pm2.43b 55	. 2 ⊢ (𝑦 ∈ On → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴))
34	33	rexlimiv 3154	1 ⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  {copab 5228   E cep 5598   We wwe 5651   × cxp 5698  ran crn 5701  Oncon0 6395  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573   ≼ cdom 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-dom 9005

This theorem is referenced by:  ondomen  10106