MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac10ct Structured version   Visualization version   GIF version

Theorem ac10ct 9934
Description: A proof of the well-ordering theorem weth 10395, 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 3441 . . . . . 6 𝑦 ∈ V
21brdom 8891 . . . . 5 (𝐴𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦)
3 f1f 6726 . . . . . . . . . . . 12 (𝑓:𝐴1-1𝑦𝑓:𝐴𝑦)
43frnd 6666 . . . . . . . . . . 11 (𝑓:𝐴1-1𝑦 → ran 𝑓𝑦)
5 onss 7726 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ⊆ On)
6 sstr2 3937 . . . . . . . . . . 11 (ran 𝑓𝑦 → (𝑦 ⊆ On → ran 𝑓 ⊆ On))
74, 5, 6syl2im 40 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → ran 𝑓 ⊆ On))
8 epweon 7716 . . . . . . . . . 10 E We On
9 wess 5607 . . . . . . . . . 10 (ran 𝑓 ⊆ On → ( E We On → E We ran 𝑓))
107, 8, 9syl6mpi 67 . . . . . . . . 9 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → E We ran 𝑓))
1110adantl 481 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → E We ran 𝑓))
12 f1f1orn 6781 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦𝑓:𝐴1-1-onto→ran 𝑓)
13 eqid 2733 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}
1413f1owe 7295 . . . . . . . . . 10 (𝑓:𝐴1-1-onto→ran 𝑓 → ( E We ran 𝑓 → {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴))
1512, 14syl 17 . . . . . . . . 9 (𝑓:𝐴1-1𝑦 → ( E We ran 𝑓 → {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴))
16 weinxp 5706 . . . . . . . . . 10 ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
17 reldom 8883 . . . . . . . . . . . 12 Rel ≼
1817brrelex1i 5677 . . . . . . . . . . 11 (𝐴𝑦𝐴 ∈ V)
19 sqxpexg 7696 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
20 incom 4158 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴))
21 inex1g 5261 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) ∈ V)
2220, 21eqeltrrid 2838 . . . . . . . . . . 11 ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
23 weeq1 5608 . . . . . . . . . . . 12 (𝑥 = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴))
2423spcegv 3548 . . . . . . . . . . 11 (({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V → (({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2518, 19, 22, 244syl 19 . . . . . . . . . 10 (𝐴𝑦 → (({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2616, 25biimtrid 242 . . . . . . . . 9 (𝐴𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2715, 26sylan9r 508 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → ( E We ran 𝑓 → ∃𝑥 𝑥 We 𝐴))
2811, 27syld 47 . . . . . . 7 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → ∃𝑥 𝑥 We 𝐴))
2928impancom 451 . . . . . 6 ((𝐴𝑦𝑦 ∈ On) → (𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
3029exlimdv 1934 . . . . 5 ((𝐴𝑦𝑦 ∈ On) → (∃𝑓 𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
312, 30biimtrid 242 . . . 4 ((𝐴𝑦𝑦 ∈ On) → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3231ex 412 . . 3 (𝐴𝑦 → (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)))
3332pm2.43b 55 . 2 (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3433rexlimiv 3127 1 (∃𝑦 ∈ On 𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1780  wcel 2113  wrex 3057  Vcvv 3437  cin 3897  wss 3898   class class class wbr 5095  {copab 5157   E cep 5520   We wwe 5573   × cxp 5619  ran crn 5622  Oncon0 6313  1-1wf1 6485  1-1-ontowf1o 6487  cfv 6488  cdom 8875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6316  df-on 6317  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-dom 8879
This theorem is referenced by:  ondomen  9937
  Copyright terms: Public domain W3C validator