Theorem ac10ct 9947

Description: A proof of the well-ordering theorem weth 10408, 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 3442	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	brdom 8893	. . . . 5 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)
3		f1f 6724	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴⟶𝑦)
4	3	frnd 6664	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1→𝑦 → ran 𝑓 ⊆ 𝑦)
5		onss 7725	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
6		sstr2 3944	. . . . . . . . . . 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 7715	. . . . . . . . . 10 ⊢ E We On
9		wess 5609	. . . . . . . . . 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 6779	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴–1-1-onto→ran 𝑓)
13		eqid 2729	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}
14	13	f1owe 7294	. . . . . . . . . 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 5708	. . . . . . . . . 10 ⊢ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
17		reldom 8885	. . . . . . . . . . . 12 ⊢ Rel ≼
18	17	brrelex1i 5679	. . . . . . . . . . 11 ⊢ (𝐴 ≼ 𝑦 → 𝐴 ∈ V)
19		sqxpexg 7695	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
20		incom 4162	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴))
21		inex1g 5261	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) ∈ V)
22	20, 21	eqeltrrid 2833	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
23		weeq1 5610	. . . . . . . . . . . 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 3554	. . . . . . . . . . 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 1933	. . . . 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 3123	1 ⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5095  {copab 5157   E cep 5522   We wwe 5575   × cxp 5621  ran crn 5624  Oncon0 6311  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486   ≼ cdom 8877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-dom 8881

This theorem is referenced by:  ondomen  9950