Theorem ac10ct 9970

Description: A proof of the well-ordering theorem weth 10431, 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 3449	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	brdom 8900	. . . . 5 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)
3		f1f 6738	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴⟶𝑦)
4	3	frnd 6676	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1→𝑦 → ran 𝑓 ⊆ 𝑦)
5		onss 7719	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
6		sstr2 3951	. . . . . . . . . . 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 7709	. . . . . . . . . 10 ⊢ E We On
9		wess 5620	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) → (𝑦 ∈ On → E We ran 𝑓))
12		f1f1orn 6795	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴–1-1-onto→ran 𝑓)
13		eqid 2736	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}
14	13	f1owe 7298	. . . . . . . . . 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 5716	. . . . . . . . . 10 ⊢ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
17		reldom 8889	. . . . . . . . . . . 12 ⊢ Rel ≼
18	17	brrelex1i 5688	. . . . . . . . . . 11 ⊢ (𝐴 ≼ 𝑦 → 𝐴 ∈ V)
19		sqxpexg 7689	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
20		incom 4161	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴))
21		inex1g 5276	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) ∈ V)
22	20, 21	eqeltrrid 2843	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
23		weeq1 5621	. . . . . . . . . . . 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 3556	. . . . . . . . . . 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 241	. . . . . . . . 9 ⊢ (𝐴 ≼ 𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
27	15, 26	sylan9r 509	. . . . . . . 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 452	. . . . . 6 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (𝑓:𝐴–1-1→𝑦 → ∃𝑥 𝑥 We 𝐴))
30	29	exlimdv 1936	. . . . 5 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (∃𝑓 𝑓:𝐴–1-1→𝑦 → ∃𝑥 𝑥 We 𝐴))
31	2, 30	biimtrid 241	. . . 4 ⊢ ((𝐴 ≼ 𝑦 ∧ 𝑦 ∈ On) → (𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴))
32	31	ex 413	. . 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 3145	1 ⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1781   ∈ wcel 2106  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910   class class class wbr 5105  {copab 5167   E cep 5536   We wwe 5587   × cxp 5631  ran crn 5634  Oncon0 6317  –1-1→wf1 6493  –1-1-onto→wf1o 6495  ‘cfv 6496   ≼ cdom 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-dom 8885

This theorem is referenced by:  ondomen  9973