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Theorem ac10ct 9057
Description: A proof of the Well ordering theorem weth 9519, 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 3354 . . . . . 6 𝑦 ∈ V
21brdom 8121 . . . . 5 (𝐴𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦)
3 f1f 6241 . . . . . . . . . . . 12 (𝑓:𝐴1-1𝑦𝑓:𝐴𝑦)
4 frn 6193 . . . . . . . . . . . 12 (𝑓:𝐴𝑦 → ran 𝑓𝑦)
53, 4syl 17 . . . . . . . . . . 11 (𝑓:𝐴1-1𝑦 → ran 𝑓𝑦)
6 onss 7137 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ⊆ On)
7 sstr2 3759 . . . . . . . . . . 11 (ran 𝑓𝑦 → (𝑦 ⊆ On → ran 𝑓 ⊆ On))
85, 6, 7syl2im 40 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → ran 𝑓 ⊆ On))
9 epweon 7130 . . . . . . . . . 10 E We On
10 wess 5236 . . . . . . . . . 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 467 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → E We ran 𝑓))
13 f1f1orn 6289 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦𝑓:𝐴1-1-onto→ran 𝑓)
14 eqid 2771 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}
1514f1owe 6746 . . . . . . . . . 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 5326 . . . . . . . . . 10 ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
18 reldom 8115 . . . . . . . . . . . 12 Rel ≼
1918brrelexi 5298 . . . . . . . . . . 11 (𝐴𝑦𝐴 ∈ V)
20 sqxpexg 7110 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
21 incom 3956 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴))
22 inex1g 4935 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) ∈ V)
2321, 22syl5eqelr 2855 . . . . . . . . . . 11 ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
24 weeq1 5237 . . . . . . . . . . . 12 (𝑥 = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴))
2524spcegv 3445 . . . . . . . . . . 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 492 . . . . . . . 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 439 . . . . . 6 ((𝐴𝑦𝑦 ∈ On) → (𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
3130exlimdv 2013 . . . . 5 ((𝐴𝑦𝑦 ∈ On) → (∃𝑓 𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
322, 31syl5bi 232 . . . 4 ((𝐴𝑦𝑦 ∈ On) → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3332ex 397 . . 3 (𝐴𝑦 → (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)))
3433pm2.43b 55 . 2 (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3534rexlimiv 3175 1 (∃𝑦 ∈ On 𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1852  wcel 2145  wrex 3062  Vcvv 3351  cin 3722  wss 3723   class class class wbr 4786  {copab 4846   E cep 5161   We wwe 5207   × cxp 5247  ran crn 5250  Oncon0 5866  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  cdom 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-dom 8111
This theorem is referenced by:  ondomen  9060
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