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Theorem cardaleph 10104
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 9959 . . . . . . . . 9 (cardβ€˜π΄) ∈ On
2 eleq1 2816 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 alephle 10103 . . . . . . . . 9 (𝐴 ∈ On β†’ 𝐴 βŠ† (β„΅β€˜π΄))
5 fveq2 6891 . . . . . . . . . . 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 686 . . . . . . . 8 (𝐴 ∈ On β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯))
9 nfcv 2898 . . . . . . . . . 10 β„²π‘₯𝐴
10 nfcv 2898 . . . . . . . . . . 11 β„²π‘₯β„΅
11 nfrab1 3446 . . . . . . . . . . . 12 β„²π‘₯{π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
1211nfint 4954 . . . . . . . . . . 11 β„²π‘₯∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
1310, 12nffv 6901 . . . . . . . . . 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 6891 . . . . . . . . . 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 7791 . . . . . . . 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 6891 . . . . . . . . 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 10081 . . . . . . . . 9 (β„΅β€˜βˆ…) = Ο‰
2220, 21eqtrdi 2783 . . . . . . . 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 608 . . . . 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 458 . 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 6891 . . . . . . . . . . 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 7796 . . . . . . . . 9 (𝑦 ∈ On β†’ (𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦)))
33 vex 3473 . . . . . . . . . . 11 𝑦 ∈ V
3433sucid 6445 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
35 eleq2 2817 . . . . . . . . . 10 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 β†’ (𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ↔ 𝑦 ∈ suc 𝑦))
3634, 35mpbiri 258 . . . . . . . . 9 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦 β†’ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
3732, 36impel 505 . . . . . . . 8 ((𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
3837adantl 481 . . . . . . 7 (((cardβ€˜π΄) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦)) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
39 fveq2 6891 . . . . . . . . . . 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 10083 . . . . . . . . . . 11 (𝑦 ∈ On β†’ (β„΅β€˜suc 𝑦) = (harβ€˜(β„΅β€˜π‘¦)))
4139, 40sylan9eqr 2789 . . . . . . . . . 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 2814 . . . . . . . . 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 9576 . . . . . . . . . 10 (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) ↔ (𝐴 ∈ On ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
4544simprbi 496 . . . . . . . . 9 (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) β†’ 𝐴 β‰Ό (β„΅β€˜π‘¦))
46 onenon 9964 . . . . . . . . . . . 12 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
473, 46syl 17 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ dom card)
48 alephon 10084 . . . . . . . . . . . 12 (β„΅β€˜π‘¦) ∈ On
49 onenon 9964 . . . . . . . . . . . 12 ((β„΅β€˜π‘¦) ∈ On β†’ (β„΅β€˜π‘¦) ∈ dom card)
5048, 49ax-mp 5 . . . . . . . . . . 11 (β„΅β€˜π‘¦) ∈ dom card
51 carddom2 9992 . . . . . . . . . . 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 585 . . . . . . . . . 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 10085 . . . . . . . . . . . 12 (cardβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)
5554sseq2i 4007 . . . . . . . . . . 11 (𝐴 βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦))
5653, 55bitrdi 287 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
5752, 56bitr3d 281 . . . . . . . . 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 508 . . . . . . 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 3151 . . . . 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 7794 . . . . . . . . . . . . . 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 6388 . . . . . . . . . . . . 13 ((∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝑦 ∈ On)
6563, 64sylan 579 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝑦 ∈ On)
6632adantld 490 . . . . . . . . . . . 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 6478 . . . . . . . . . . 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 3144 . . . . . . . . 9 (𝐴 ∈ On β†’ Β¬ βˆƒπ‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}𝐴 ∈ (β„΅β€˜π‘¦))
7170adantr 480 . . . . . . . 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 10082 . . . . . . . . . . 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 579 . . . . . . . . . 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 2814 . . . . . . . . 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 4995 . . . . . . . . 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 287 . . . . . . . 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 325 . . . . . . 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 412 . . . . . 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 858 . . . 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 10084 . . . . . . 7 (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∈ On
83 onsseleq 6404 . . . . . . 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 690 . . . . . 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 863 . . . 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 481 . 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 6373 . . . . 5 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
90 ordzsl 7843 . . . . . 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 1088 . . . . . 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 275 . . . . 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 481 . 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 859 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 395   ∨ wo 846   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065  {crab 3427   βŠ† wss 3944  βˆ…c0 4318  βˆ© cint 4944  βˆͺ ciun 4991   class class class wbr 5142  dom cdm 5672  Ord word 6362  Oncon0 6363  Lim wlim 6364  suc csuc 6365  β€˜cfv 6542  Ο‰com 7864   β‰Ό cdom 8953  harchar 9571  cardccrd 9950  β„΅cale 9951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9525  df-har 9572  df-card 9954  df-aleph 9955
This theorem is referenced by:  cardalephex  10105  tskcard  10796  minregex  42887
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