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Theorem cardaleph 10112
Description: Given any transfinite cardinal number 𝐴, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
cardaleph ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
Distinct variable group:   π‘₯,𝐴
Proof of Theorem cardaleph
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardon 9967 . . . . . . . . 9 (cardβ€˜π΄) ∈ On
2 eleq1 2813 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 alephle 10111 . . . . . . . . 9 (𝐴 ∈ On β†’ 𝐴 βŠ† (β„΅β€˜π΄))
5 fveq2 6894 . . . . . . . . . . 11 (π‘₯ = 𝐴 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜π΄))
65sseq2d 4010 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (𝐴 βŠ† (β„΅β€˜π‘₯) ↔ 𝐴 βŠ† (β„΅β€˜π΄)))
76rspcev 3607 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐴 βŠ† (β„΅β€˜π΄)) β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯))
84, 7mpdan 685 . . . . . . . 8 (𝐴 ∈ On β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯))
9 nfcv 2892 . . . . . . . . . 10 β„²π‘₯𝐴
10 nfcv 2892 . . . . . . . . . . 11 β„²π‘₯β„΅
11 nfrab1 3439 . . . . . . . . . . . 12 β„²π‘₯{π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
1211nfint 4959 . . . . . . . . . . 11 β„²π‘₯∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
1310, 12nffv 6904 . . . . . . . . . 10 β„²π‘₯(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
149, 13nfss 3970 . . . . . . . . 9 β„²π‘₯ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
15 fveq2 6894 . . . . . . . . . 10 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ (β„΅β€˜π‘₯) = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
1615sseq2d 4010 . . . . . . . . 9 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ (𝐴 βŠ† (β„΅β€˜π‘₯) ↔ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
1714, 16onminsb 7796 . . . . . . . 8 (βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯) β†’ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
183, 8, 173syl 18 . . . . . . 7 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
1918a1i 11 . . . . . 6 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
20 fveq2 6894 . . . . . . . . 9 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = (β„΅β€˜βˆ…))
21 aleph0 10089 . . . . . . . . 9 (β„΅β€˜βˆ…) = Ο‰
2220, 21eqtrdi 2781 . . . . . . . 8 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = Ο‰)
2322sseq1d 4009 . . . . . . 7 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ ((β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† 𝐴 ↔ Ο‰ βŠ† 𝐴))
2423biimprd 247 . . . . . 6 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ (Ο‰ βŠ† 𝐴 β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† 𝐴))
2519, 24anim12d 607 . . . . 5 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) β†’ (𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† 𝐴)))
26 eqss 3993 . . . . 5 (𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ (𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† 𝐴))
2725, 26imbitrrdi 251 . . . 4 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
2827com12 32 . . 3 (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
2928ancoms 457 . 2 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
30 fveq2 6894 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜π‘¦))
3130sseq2d 4010 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ (𝐴 βŠ† (β„΅β€˜π‘₯) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
3231onnminsb 7801 . . . . . . . . 9 (𝑦 ∈ On β†’ (𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦)))
33 vex 3467 . . . . . . . . . . 11 𝑦 ∈ V
3433sucid 6451 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
35 eleq2 2814 . . . . . . . . . 10 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 β†’ (𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ↔ 𝑦 ∈ suc 𝑦))
3634, 35mpbiri 257 . . . . . . . . 9 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 β†’ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
3732, 36impel 504 . . . . . . . 8 ((𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
3837adantl 480 . . . . . . 7 (((cardβ€˜π΄) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦)) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
39 fveq2 6894 . . . . . . . . . . 11 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = (β„΅β€˜suc 𝑦))
40 alephsuc 10091 . . . . . . . . . . 11 (𝑦 ∈ On β†’ (β„΅β€˜suc 𝑦) = (harβ€˜(β„΅β€˜π‘¦)))
4139, 40sylan9eqr 2787 . . . . . . . . . 10 ((𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦) β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = (harβ€˜(β„΅β€˜π‘¦)))
4241eleq2d 2811 . . . . . . . . 9 ((𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦) β†’ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ 𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦))))
4342biimpd 228 . . . . . . . 8 ((𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦) β†’ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦))))
44 elharval 9584 . . . . . . . . . 10 (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) ↔ (𝐴 ∈ On ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
4544simprbi 495 . . . . . . . . 9 (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) β†’ 𝐴 β‰Ό (β„΅β€˜π‘¦))
46 onenon 9972 . . . . . . . . . . . 12 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
473, 46syl 17 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ dom card)
48 alephon 10092 . . . . . . . . . . . 12 (β„΅β€˜π‘¦) ∈ On
49 onenon 9972 . . . . . . . . . . . 12 ((β„΅β€˜π‘¦) ∈ On β†’ (β„΅β€˜π‘¦) ∈ dom card)
5048, 49ax-mp 5 . . . . . . . . . . 11 (β„΅β€˜π‘¦) ∈ dom card
51 carddom2 10000 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ (β„΅β€˜π‘¦) ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
5247, 50, 51sylancl 584 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
53 sseq1 4003 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (cardβ€˜(β„΅β€˜π‘¦))))
54 alephcard 10093 . . . . . . . . . . . 12 (cardβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)
5554sseq2i 4007 . . . . . . . . . . 11 (𝐴 βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦))
5653, 55bitrdi 286 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
5752, 56bitr3d 280 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ (𝐴 β‰Ό (β„΅β€˜π‘¦) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
5845, 57imbitrid 243 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) β†’ 𝐴 βŠ† (β„΅β€˜π‘¦)))
5943, 58sylan9r 507 . . . . . . 7 (((cardβ€˜π΄) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦)) β†’ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝐴 βŠ† (β„΅β€˜π‘¦)))
6038, 59mtod 197 . . . . . 6 (((cardβ€˜π΄) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦)) β†’ Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
6160rexlimdvaa 3146 . . . . 5 ((cardβ€˜π΄) = 𝐴 β†’ (βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 β†’ Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
62 onintrab2 7799 . . . . . . . . . . . . . 14 (βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯) ↔ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
638, 62sylib 217 . . . . . . . . . . . . 13 (𝐴 ∈ On β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
64 onelon 6394 . . . . . . . . . . . . 13 ((∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝑦 ∈ On)
6563, 64sylan 578 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝑦 ∈ On)
6632adantld 489 . . . . . . . . . . . 12 (𝑦 ∈ On β†’ ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦)))
6765, 66mpcom 38 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
6848onelssi 6484 . . . . . . . . . . 11 (𝐴 ∈ (β„΅β€˜π‘¦) β†’ 𝐴 βŠ† (β„΅β€˜π‘¦))
6967, 68nsyl 140 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ Β¬ 𝐴 ∈ (β„΅β€˜π‘¦))
7069nrexdv 3139 . . . . . . . . 9 (𝐴 ∈ On β†’ Β¬ βˆƒπ‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}𝐴 ∈ (β„΅β€˜π‘¦))
7170adantr 479 . . . . . . . 8 ((𝐴 ∈ On ∧ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ Β¬ βˆƒπ‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}𝐴 ∈ (β„΅β€˜π‘¦))
72 alephlim 10090 . . . . . . . . . . 11 ((∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = βˆͺ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} (β„΅β€˜π‘¦))
7363, 72sylan 578 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = βˆͺ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} (β„΅β€˜π‘¦))
7473eleq2d 2811 . . . . . . . . 9 ((𝐴 ∈ On ∧ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ 𝐴 ∈ βˆͺ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} (β„΅β€˜π‘¦)))
75 eliun 5000 . . . . . . . . 9 (𝐴 ∈ βˆͺ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} (β„΅β€˜π‘¦) ↔ βˆƒπ‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}𝐴 ∈ (β„΅β€˜π‘¦))
7674, 75bitrdi 286 . . . . . . . 8 ((𝐴 ∈ On ∧ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ βˆƒπ‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}𝐴 ∈ (β„΅β€˜π‘¦)))
7771, 76mtbird 324 . . . . . . 7 ((𝐴 ∈ On ∧ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
7877ex 411 . . . . . 6 (𝐴 ∈ On β†’ (Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
793, 78syl 17 . . . . 5 ((cardβ€˜π΄) = 𝐴 β†’ (Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
8061, 79jaod 857 . . . 4 ((cardβ€˜π΄) = 𝐴 β†’ ((βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
818, 17syl 17 . . . . . 6 (𝐴 ∈ On β†’ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
82 alephon 10092 . . . . . . 7 (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∈ On
83 onsseleq 6410 . . . . . . 7 ((𝐴 ∈ On ∧ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∈ On) β†’ (𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∨ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))))
8482, 83mpan2 689 . . . . . 6 (𝐴 ∈ On β†’ (𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∨ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))))
8581, 84mpbid 231 . . . . 5 (𝐴 ∈ On β†’ (𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∨ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
8685ord 862 . . . 4 (𝐴 ∈ On β†’ (Β¬ 𝐴 ∈ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
873, 80, 86sylsyld 61 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ ((βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
8887adantl 480 . 2 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ ((βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
89 eloni 6379 . . . . 5 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
90 ordzsl 7848 . . . . . 6 (Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ↔ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
91 3orass 1087 . . . . . 6 ((∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ↔ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ (βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
9290, 91bitri 274 . . . . 5 (Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ↔ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ (βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
9389, 92sylib 217 . . . 4 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ (βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
943, 63, 933syl 18 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ (βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
9594adantl 480 . 2 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… ∨ (βˆƒπ‘¦ ∈ On ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 ∨ Lim ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
9629, 88, 95mpjaod 858 1 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060  {crab 3419   βŠ† wss 3945  βˆ…c0 4323  βˆ© cint 4949  βˆͺ ciun 4996   class class class wbr 5148  dom cdm 5677  Ord word 6368  Oncon0 6369  Lim wlim 6370  suc csuc 6371  β€˜cfv 6547  Ο‰com 7869   β‰Ό cdom 8960  harchar 9579  cardccrd 9958  β„΅cale 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  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-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7373  df-ov 7420  df-om 7870  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-har 9580  df-card 9962  df-aleph 9963
This theorem is referenced by:  cardalephex  10113  tskcard  10804  minregex  43029
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