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Theorem ac10ct 9508
 Description: A proof of the well-ordering theorem weth 9969, 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 3414 . . . . . 6 𝑦 ∈ V
21brdom 8553 . . . . 5 (𝐴𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦)
3 f1f 6566 . . . . . . . . . . . 12 (𝑓:𝐴1-1𝑦𝑓:𝐴𝑦)
43frnd 6511 . . . . . . . . . . 11 (𝑓:𝐴1-1𝑦 → ran 𝑓𝑦)
5 onss 7511 . . . . . . . . . . 11 (𝑦 ∈ On → 𝑦 ⊆ On)
6 sstr2 3902 . . . . . . . . . . 11 (ran 𝑓𝑦 → (𝑦 ⊆ On → ran 𝑓 ⊆ On))
74, 5, 6syl2im 40 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦 → (𝑦 ∈ On → ran 𝑓 ⊆ On))
8 epweon 7503 . . . . . . . . . 10 E We On
9 wess 5516 . . . . . . . . . 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 485 . . . . . . . 8 ((𝐴𝑦𝑓:𝐴1-1𝑦) → (𝑦 ∈ On → E We ran 𝑓))
12 f1f1orn 6619 . . . . . . . . . 10 (𝑓:𝐴1-1𝑦𝑓:𝐴1-1-onto→ran 𝑓)
13 eqid 2759 . . . . . . . . . . 11 {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} = {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}
1413f1owe 7107 . . . . . . . . . 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 5611 . . . . . . . . . 10 ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴)
17 reldom 8547 . . . . . . . . . . . 12 Rel ≼
1817brrelex1i 5583 . . . . . . . . . . 11 (𝐴𝑦𝐴 ∈ V)
19 sqxpexg 7483 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × 𝐴) ∈ V)
20 incom 4109 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴))
21 inex1g 5194 . . . . . . . . . . . 12 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)}) ∈ V)
2220, 21eqeltrrid 2858 . . . . . . . . . . 11 ((𝐴 × 𝐴) ∈ V → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) ∈ V)
23 weeq1 5517 . . . . . . . . . . . 12 (𝑥 = ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} ∩ (𝐴 × 𝐴)) We 𝐴))
2423spcegv 3518 . . . . . . . . . . 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, 25syl5bi 245 . . . . . . . . 9 (𝐴𝑦 → ({⟨𝑤, 𝑧⟩ ∣ (𝑓𝑤) E (𝑓𝑧)} We 𝐴 → ∃𝑥 𝑥 We 𝐴))
2715, 26sylan9r 512 . . . . . . . 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 455 . . . . . 6 ((𝐴𝑦𝑦 ∈ On) → (𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
3029exlimdv 1935 . . . . 5 ((𝐴𝑦𝑦 ∈ On) → (∃𝑓 𝑓:𝐴1-1𝑦 → ∃𝑥 𝑥 We 𝐴))
312, 30syl5bi 245 . . . 4 ((𝐴𝑦𝑦 ∈ On) → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3231ex 416 . . 3 (𝐴𝑦 → (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)))
3332pm2.43b 55 . 2 (𝑦 ∈ On → (𝐴𝑦 → ∃𝑥 𝑥 We 𝐴))
3433rexlimiv 3205 1 (∃𝑦 ∈ On 𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1782   ∈ wcel 2112  ∃wrex 3072  Vcvv 3410   ∩ cin 3860   ⊆ wss 3861   class class class wbr 5037  {copab 5099   E cep 5439   We wwe 5487   × cxp 5527  ran crn 5530  Oncon0 6175  –1-1→wf1 6338  –1-1-onto→wf1o 6340  ‘cfv 6341   ≼ cdom 8539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-ord 6178  df-on 6179  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-isom 6350  df-dom 8543 This theorem is referenced by:  ondomen  9511
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