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Theorem ac10ct 8895
Description: A proof of the Well ordering theorem weth 9355, 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 𝐴𝑦 → ∃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.
StepHypRef Expression
1 vex 3234 . . . . . 6 𝑦 ∈ V
21brdom 8009 . . . . 5 (𝐴𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦)
3 f1f 6139 . . . . . . . . . . . 12 (𝑓:𝐴1-1𝑦𝑓:𝐴𝑦)
4 frn 6091 . . . . . . . . . . . 12 (𝑓:𝐴𝑦 → ran 𝑓𝑦)
53, 4syl 17 . . . . . . . . . . 11 (𝑓:𝐴1-1𝑦 → ran 𝑓𝑦)
6 onss 7032 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ⊆ On)
7 sstr2 3643 . . . . . . . . . . 11 (ran 𝑓𝑦 → (𝑦 ⊆ On → ran 𝑓 ⊆ On))
85, 6, 7syl2im 40 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → ran 𝑓 ⊆ On))
9 epweon 7025 . . . . . . . . . 10 E We On
10 wess 5130 . . . . . . . . . 10 (ran 𝑓 ⊆ On → ( E We On → E We ran 𝑓))
118, 9, 10syl6mpi 67 . . . . . . . . 9 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → E We ran 𝑓))
1211adantl 481 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → E We ran 𝑓))
13 f1f1orn 6186 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦𝑓:𝐴1-1-onto→ran 𝑓)
14 eqid 2651 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}
1514f1owe 6643 . . . . . . . . . 10 (𝑓:𝐴1-1-onto→ran 𝑓 → ( E We ran 𝑓 → {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴))
1613, 15syl 17 . . . . . . . . 9 (𝑓:𝐴1-1𝑦 → ( E We ran 𝑓 → {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴))
17 weinxp 5220 . . . . . . . . . 10 ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
18 reldom 8003 . . . . . . . . . . . 12 Rel ≼
1918brrelexi 5192 . . . . . . . . . . 11 (𝐴𝑦𝐴 ∈ V)
20 sqxpexg 7005 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
21 incom 3838 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴))
22 inex1g 4834 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) ∈ V)
2321, 22syl5eqelr 2735 . . . . . . . . . . 11 ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
24 weeq1 5131 . . . . . . . . . . . 12 (𝑥 = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴))
2524spcegv 3325 . . . . . . . . . . 11 (({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V → (({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2619, 20, 23, 254syl 19 . . . . . . . . . 10 (𝐴𝑦 → (({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2717, 26syl5bi 232 . . . . . . . . 9 (𝐴𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2816, 27sylan9r 691 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → ( E We ran 𝑓 → ∃𝑥 𝑥 We 𝐴))
2912, 28syld 47 . . . . . . 7 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → ∃𝑥 𝑥 We 𝐴))
3029impancom 455 . . . . . 6 ((𝐴𝑦𝑦 ∈ On) → (𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
3130exlimdv 1901 . . . . 5 ((𝐴𝑦𝑦 ∈ On) → (∃𝑓 𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
322, 31syl5bi 232 . . . 4 ((𝐴𝑦𝑦 ∈ On) → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3332ex 449 . . 3 (𝐴𝑦 → (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)))
3433pm2.43b 55 . 2 (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3534rexlimiv 3056 1 (∃𝑦 ∈ On 𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  cin 3606  wss 3607   class class class wbr 4685  {copab 4745   E cep 5057   We wwe 5101   × cxp 5141  ran crn 5144  Oncon0 5761  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-dom 7999
This theorem is referenced by:  ondomen  8898
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