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Theorem cardaleph 10084
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 9939 . . . . . . . . 9 (cardβ€˜π΄) ∈ On
2 eleq1 2822 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 232 . . . . . . . 8 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
4 alephle 10083 . . . . . . . . 9 (𝐴 ∈ On β†’ 𝐴 βŠ† (β„΅β€˜π΄))
5 fveq2 6892 . . . . . . . . . . 11 (π‘₯ = 𝐴 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜π΄))
65sseq2d 4015 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (𝐴 βŠ† (β„΅β€˜π‘₯) ↔ 𝐴 βŠ† (β„΅β€˜π΄)))
76rspcev 3613 . . . . . . . . 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 2904 . . . . . . . . . 10 β„²π‘₯𝐴
10 nfcv 2904 . . . . . . . . . . 11 β„²π‘₯β„΅
11 nfrab1 3452 . . . . . . . . . . . 12 β„²π‘₯{π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
1211nfint 4961 . . . . . . . . . . 11 β„²π‘₯∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
1310, 12nffv 6902 . . . . . . . . . 10 β„²π‘₯(β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
149, 13nfss 3975 . . . . . . . . 9 β„²π‘₯ 𝐴 βŠ† (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
15 fveq2 6892 . . . . . . . . . 10 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ (β„΅β€˜π‘₯) = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
1615sseq2d 4015 . . . . . . . . 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 7782 . . . . . . . 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 6892 . . . . . . . . 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 10061 . . . . . . . . 9 (β„΅β€˜βˆ…) = Ο‰
2220, 21eqtrdi 2789 . . . . . . . 8 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = βˆ… β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) = Ο‰)
2322sseq1d 4014 . . . . . . 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 610 . . . . 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 3998 . . . . 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 460 . 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 6892 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (β„΅β€˜π‘₯) = (β„΅β€˜π‘¦))
3130sseq2d 4015 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ (𝐴 βŠ† (β„΅β€˜π‘₯) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
3231onnminsb 7787 . . . . . . . . 9 (𝑦 ∈ On β†’ (𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦)))
33 vex 3479 . . . . . . . . . . 11 𝑦 ∈ V
3433sucid 6447 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
35 eleq2 2823 . . . . . . . . . 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 507 . . . . . . . 8 ((𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
3837adantl 483 . . . . . . 7 (((cardβ€˜π΄) = 𝐴 ∧ (𝑦 ∈ On ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} = suc 𝑦)) β†’ Β¬ 𝐴 βŠ† (β„΅β€˜π‘¦))
39 fveq2 6892 . . . . . . . . . . 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 10063 . . . . . . . . . . 11 (𝑦 ∈ On β†’ (β„΅β€˜suc 𝑦) = (harβ€˜(β„΅β€˜π‘¦)))
4139, 40sylan9eqr 2795 . . . . . . . . . 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 2820 . . . . . . . . 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 9556 . . . . . . . . . 10 (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) ↔ (𝐴 ∈ On ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
4544simprbi 498 . . . . . . . . 9 (𝐴 ∈ (harβ€˜(β„΅β€˜π‘¦)) β†’ 𝐴 β‰Ό (β„΅β€˜π‘¦))
46 onenon 9944 . . . . . . . . . . . 12 (𝐴 ∈ On β†’ 𝐴 ∈ dom card)
473, 46syl 17 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ dom card)
48 alephon 10064 . . . . . . . . . . . 12 (β„΅β€˜π‘¦) ∈ On
49 onenon 9944 . . . . . . . . . . . 12 ((β„΅β€˜π‘¦) ∈ On β†’ (β„΅β€˜π‘¦) ∈ dom card)
5048, 49ax-mp 5 . . . . . . . . . . 11 (β„΅β€˜π‘¦) ∈ dom card
51 carddom2 9972 . . . . . . . . . . 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 587 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
53 sseq1 4008 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (cardβ€˜(β„΅β€˜π‘¦))))
54 alephcard 10065 . . . . . . . . . . . 12 (cardβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)
5554sseq2i 4012 . . . . . . . . . . 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 510 . . . . . . 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 3157 . . . . 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 7785 . . . . . . . . . . . . . 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 6390 . . . . . . . . . . . . 13 ((∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝑦 ∈ On)
6563, 64sylan 581 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) β†’ 𝑦 ∈ On)
6632adantld 492 . . . . . . . . . . . 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 6480 . . . . . . . . . . 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 3150 . . . . . . . . 9 (𝐴 ∈ On β†’ Β¬ βˆƒπ‘¦ ∈ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}𝐴 ∈ (β„΅β€˜π‘¦))
7170adantr 482 . . . . . . . 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 10062 . . . . . . . . . . 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 581 . . . . . . . . . 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 2820 . . . . . . . . 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 5002 . . . . . . . . 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 414 . . . . . 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 10064 . . . . . . 7 (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∈ On
83 onsseleq 6406 . . . . . . 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 483 . 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 6375 . . . . 5 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
90 ordzsl 7834 . . . . . 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 1091 . . . . . 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 483 . 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 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433   βŠ† wss 3949  βˆ…c0 4323  βˆ© cint 4951  βˆͺ ciun 4998   class class class wbr 5149  dom cdm 5677  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  β€˜cfv 6544  Ο‰com 7855   β‰Ό cdom 8937  harchar 9551  cardccrd 9930  β„΅cale 9931
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  ax-inf2 9636
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-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  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-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-har 9552  df-card 9934  df-aleph 9935
This theorem is referenced by:  cardalephex  10085  tskcard  10776  minregex  42285
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