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 3434	. . . . . 6 ⊢ 𝑦 ∈ V
2	1	brdom 8900	. . . . 5 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)
3		f1f 6730	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴⟶𝑦)
4	3	frnd 6670	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1→𝑦 → ran 𝑓 ⊆ 𝑦)
5		onss 7732	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → 𝑦 ⊆ On)
6		sstr2 3929	. . . . . . . . . . 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 7722	. . . . . . . . . 10 ⊢ E We On
9		wess 5610	. . . . . . . . . 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 6785	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝑦 → 𝑓:𝐴–1-1-onto→ran 𝑓)
13		eqid 2737	. . . . . . . . . . 11 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}
14	13	f1owe 7301	. . . . . . . . . 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 5709	. . . . . . . . . 10 ⊢ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
17		reldom 8892	. . . . . . . . . . . 12 ⊢ Rel ≼
18	17	brrelex1i 5680	. . . . . . . . . . 11 ⊢ (𝐴 ≼ 𝑦 → 𝐴 ∈ V)
19		sqxpexg 7702	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
20		incom 4150	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴))
21		inex1g 5256	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)}) ∈ V)
22	20, 21	eqeltrrid 2842	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓‘𝑤) E (𝑓‘𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
23		weeq1 5611	. . . . . . . . . . . 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 3540	. . . . . . . . . . 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 1935	. . . . 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 3132	1 ⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890   class class class wbr 5086  {copab 5148   E cep 5523   We wwe 5576   × cxp 5622  ran crn 5625  Oncon0 6317  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492   ≼ cdom 8884

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-dom 8888

This theorem is referenced by:  ondomen  9950