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Theorem ac10ct 10031
Description: A proof of the well-ordering theorem weth 10492, 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 3478 . . . . . 6 𝑦 ∈ V
21brdom 8958 . . . . 5 (𝐴 β‰Ό 𝑦 ↔ βˆƒπ‘“ 𝑓:𝐴–1-1→𝑦)
3 f1f 6787 . . . . . . . . . . . 12 (𝑓:𝐴–1-1→𝑦 β†’ 𝑓:π΄βŸΆπ‘¦)
43frnd 6725 . . . . . . . . . . 11 (𝑓:𝐴–1-1→𝑦 β†’ ran 𝑓 βŠ† 𝑦)
5 onss 7774 . . . . . . . . . . 11 (𝑦 ∈ On β†’ 𝑦 βŠ† On)
6 sstr2 3989 . . . . . . . . . . 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 7764 . . . . . . . . . 10 E We On
9 wess 5663 . . . . . . . . . 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 482 . . . . . . . 8 ((𝐴 β‰Ό 𝑦 ∧ 𝑓:𝐴–1-1→𝑦) β†’ (𝑦 ∈ On β†’ E We ran 𝑓))
12 f1f1orn 6844 . . . . . . . . . 10 (𝑓:𝐴–1-1→𝑦 β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
13 eqid 2732 . . . . . . . . . . 11 {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} = {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)}
1413f1owe 7352 . . . . . . . . . 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 5760 . . . . . . . . . 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 8947 . . . . . . . . . . . 12 Rel β‰Ό
1817brrelex1i 5732 . . . . . . . . . . 11 (𝐴 β‰Ό 𝑦 β†’ 𝐴 ∈ V)
19 sqxpexg 7744 . . . . . . . . . . 11 (𝐴 ∈ V β†’ (𝐴 Γ— 𝐴) ∈ V)
20 incom 4201 . . . . . . . . . . . 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 5319 . . . . . . . . . . . 12 ((𝐴 Γ— 𝐴) ∈ V β†’ ((𝐴 Γ— 𝐴) ∩ {βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)}) ∈ V)
2220, 21eqeltrrid 2838 . . . . . . . . . . 11 ((𝐴 Γ— 𝐴) ∈ V β†’ ({βŸ¨π‘€, π‘§βŸ© ∣ (π‘“β€˜π‘€) E (π‘“β€˜π‘§)} ∩ (𝐴 Γ— 𝐴)) ∈ V)
23 weeq1 5664 . . . . . . . . . . . 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 3587 . . . . . . . . . . 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 509 . . . . . . . 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 452 . . . . . 6 ((𝐴 β‰Ό 𝑦 ∧ 𝑦 ∈ On) β†’ (𝑓:𝐴–1-1→𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴))
3029exlimdv 1936 . . . . 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 413 . . 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 3148 1 (βˆƒπ‘¦ ∈ On 𝐴 β‰Ό 𝑦 β†’ βˆƒπ‘₯ π‘₯ We 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396  βˆƒwex 1781   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948   class class class wbr 5148  {copab 5210   E cep 5579   We wwe 5630   Γ— cxp 5674  ran crn 5677  Oncon0 6364  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543   β‰Ό cdom 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-dom 8943
This theorem is referenced by:  ondomen  10034
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