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Theorem ac10ct 8740
 Description: A proof of the Well ordering theorem weth 9200, 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 3176 . . . . . 6 𝑦 ∈ V
21brdom 7853 . . . . 5 (𝐴𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦)
3 f1f 6014 . . . . . . . . . . . 12 (𝑓:𝐴1-1𝑦𝑓:𝐴𝑦)
4 frn 5966 . . . . . . . . . . . 12 (𝑓:𝐴𝑦 → ran 𝑓𝑦)
53, 4syl 17 . . . . . . . . . . 11 (𝑓:𝐴1-1𝑦 → ran 𝑓𝑦)
6 onss 6882 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ⊆ On)
7 sstr2 3575 . . . . . . . . . . 11 (ran 𝑓𝑦 → (𝑦 ⊆ On → ran 𝑓 ⊆ On))
85, 6, 7syl2im 39 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → ran 𝑓 ⊆ On))
9 epweon 6875 . . . . . . . . . 10 E We On
10 wess 5025 . . . . . . . . . 10 (ran 𝑓 ⊆ On → ( E We On → E We ran 𝑓))
118, 9, 10syl6mpi 65 . . . . . . . . 9 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → E We ran 𝑓))
1211adantl 481 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → E We ran 𝑓))
13 f1f1orn 6061 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦𝑓:𝐴1-1-onto→ran 𝑓)
14 eqid 2610 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}
1514f1owe 6503 . . . . . . . . . 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 5109 . . . . . . . . . 10 ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
18 reldom 7847 . . . . . . . . . . . 12 Rel ≼
1918brrelexi 5082 . . . . . . . . . . 11 (𝐴𝑦𝐴 ∈ V)
20 sqxpexg 6861 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
21 incom 3767 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴))
22 inex1g 4729 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) ∈ V)
2321, 22syl5eqelr 2693 . . . . . . . . . . 11 ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
24 weeq1 5026 . . . . . . . . . . . 12 (𝑥 = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴))
2524spcegv 3267 . . . . . . . . . . 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 231 . . . . . . . . 9 (𝐴𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2816, 27sylan9r 688 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → ( E We ran 𝑓 → ∃𝑥 𝑥 We 𝐴))
2912, 28syld 46 . . . . . . 7 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → ∃𝑥 𝑥 We 𝐴))
3029impancom 455 . . . . . 6 ((𝐴𝑦𝑦 ∈ On) → (𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
3130exlimdv 1848 . . . . 5 ((𝐴𝑦𝑦 ∈ On) → (∃𝑓 𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
322, 31syl5bi 231 . . . 4 ((𝐴𝑦𝑦 ∈ On) → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3332ex 449 . . 3 (𝐴𝑦 → (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)))
3433pm2.43b 53 . 2 (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3534rexlimiv 3009 1 (∃𝑦 ∈ On 𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  {copab 4642   E cep 4947   We wwe 4996   × cxp 5036  ran crn 5039  Oncon0 5640  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   ≼ cdom 7839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-dom 7843 This theorem is referenced by:  ondomen  8743
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