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Theorem ac10ct 10029
Description: A proof of the well-ordering theorem weth 10490, 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 (βˆƒπ‘¦ ∈ On 𝐴 β‰Ό 𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴)
Distinct variable group:   π‘₯,𝐴,𝑦

Proof of Theorem ac10ct
Dummy variables 𝑓 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3479 . . . . . 6 𝑦 ∈ V
21brdom 8956 . . . . 5 (𝐴 β‰Ό 𝑦 ↔ βˆƒπ‘“ 𝑓:𝐴–1-1→𝑦)
3 f1f 6788 . . . . . . . . . . . 12 (𝑓:𝐴–1-1→𝑦 β†’ 𝑓:π΄βŸΆπ‘¦)
43frnd 6726 . . . . . . . . . . 11 (𝑓:𝐴–1-1→𝑦 β†’ ran 𝑓 βŠ† 𝑦)
5 onss 7772 . . . . . . . . . . 11 (𝑦 ∈ On β†’ 𝑦 βŠ† On)
6 sstr2 3990 . . . . . . . . . . 11 (ran 𝑓 βŠ† 𝑦 β†’ (𝑦 βŠ† On β†’ ran 𝑓 βŠ† On))
74, 5, 6syl2im 40 . . . . . . . . . 10 (𝑓:𝐴–1-1→𝑦 β†’ (𝑦 ∈ On β†’ ran 𝑓 βŠ† On))
8 epweon 7762 . . . . . . . . . 10 E We On
9 wess 5664 . . . . . . . . . 10 (ran 𝑓 βŠ† On β†’ ( E We On β†’ E We ran 𝑓))
107, 8, 9syl6mpi 67 . . . . . . . . 9 (𝑓:𝐴–1-1→𝑦 β†’ (𝑦 ∈ On β†’ E We ran 𝑓))
1110adantl 483 . . . . . . . 8 ((𝐴 β‰Ό 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) β†’ (𝑦 ∈ On β†’ E We ran 𝑓))
12 f1f1orn 6845 . . . . . . . . . 10 (𝑓:𝐴–1-1→𝑦 β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
13 eqid 2733 . . . . . . . . . . 11 {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} = {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)}
1413f1owe 7350 . . . . . . . . . 10 (𝑓:𝐴–1-1-ontoβ†’ran 𝑓 β†’ ( E We ran 𝑓 β†’ {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} We 𝐴))
1512, 14syl 17 . . . . . . . . 9 (𝑓:𝐴–1-1→𝑦 β†’ ( E We ran 𝑓 β†’ {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} We 𝐴))
16 weinxp 5761 . . . . . . . . . 10 ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} We 𝐴 ↔ ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) We 𝐴)
17 reldom 8945 . . . . . . . . . . . 12 Rel β‰Ό
1817brrelex1i 5733 . . . . . . . . . . 11 (𝐴 β‰Ό 𝑦 β†’ 𝐴 ∈ V)
19 sqxpexg 7742 . . . . . . . . . . 11 (𝐴 ∈ V β†’ (𝐴 Γ— 𝐴) ∈ V)
20 incom 4202 . . . . . . . . . . . 12 ((𝐴 Γ— 𝐴) ∩ {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)}) = ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴))
21 inex1g 5320 . . . . . . . . . . . 12 ((𝐴 Γ— 𝐴) ∈ V β†’ ((𝐴 Γ— 𝐴) ∩ {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)}) ∈ V)
2220, 21eqeltrrid 2839 . . . . . . . . . . 11 ((𝐴 Γ— 𝐴) ∈ V β†’ ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) ∈ V)
23 weeq1 5665 . . . . . . . . . . . 12 (π‘₯ = ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) β†’ (π‘₯ We 𝐴 ↔ ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) We 𝐴))
2423spcegv 3588 . . . . . . . . . . 11 (({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) ∈ V β†’ (({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) We 𝐴 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
2518, 19, 22, 244syl 19 . . . . . . . . . 10 (𝐴 β‰Ό 𝑦 β†’ (({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) We 𝐴 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
2616, 25biimtrid 241 . . . . . . . . 9 (𝐴 β‰Ό 𝑦 β†’ ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} We 𝐴 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
2715, 26sylan9r 510 . . . . . . . 8 ((𝐴 β‰Ό 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) β†’ ( E We ran 𝑓 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
2811, 27syld 47 . . . . . . 7 ((𝐴 β‰Ό 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) β†’ (𝑦 ∈ On β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
2928impancom 453 . . . . . 6 ((𝐴 β‰Ό 𝑦 ∧ 𝑦 ∈ On) β†’ (𝑓:𝐴–1-1→𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
3029exlimdv 1937 . . . . 5 ((𝐴 β‰Ό 𝑦 ∧ 𝑦 ∈ On) β†’ (βˆƒπ‘“ 𝑓:𝐴–1-1→𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
312, 30biimtrid 241 . . . 4 ((𝐴 β‰Ό 𝑦 ∧ 𝑦 ∈ On) β†’ (𝐴 β‰Ό 𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
3231ex 414 . . 3 (𝐴 β‰Ό 𝑦 β†’ (𝑦 ∈ On β†’ (𝐴 β‰Ό 𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴)))
3332pm2.43b 55 . 2 (𝑦 ∈ On β†’ (𝐴 β‰Ό 𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
3433rexlimiv 3149 1 (βˆƒπ‘¦ ∈ On 𝐴 β‰Ό 𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949   class class class wbr 5149  {copab 5211   E cep 5580   We wwe 5631   Γ— cxp 5675  ran crn 5678  Oncon0 6365  β€“1-1β†’wf1 6541  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544   β‰Ό cdom 8937
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-dom 8941
This theorem is referenced by:  ondomen  10032
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