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Theorem cardaleph 10126
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 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardaleph
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardon 9981 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2826 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 233 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 alephle 10125 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
5 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
65sseq2d 4027 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
76rspcev 3621 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐴 ⊆ (ℵ‘𝐴)) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
84, 7mpdan 687 . . . . . . . 8 (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
9 nfcv 2902 . . . . . . . . . 10 𝑥𝐴
10 nfcv 2902 . . . . . . . . . . 11 𝑥
11 nfrab1 3453 . . . . . . . . . . . 12 𝑥{𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1211nfint 4960 . . . . . . . . . . 11 𝑥 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1310, 12nffv 6916 . . . . . . . . . 10 𝑥(ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
149, 13nfss 3987 . . . . . . . . 9 𝑥 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
15 fveq2 6906 . . . . . . . . . 10 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (ℵ‘𝑥) = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
1615sseq2d 4027 . . . . . . . . 9 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
1714, 16onminsb 7813 . . . . . . . 8 (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) → 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
183, 8, 173syl 18 . . . . . . 7 ((card‘𝐴) = 𝐴𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
1918a1i 11 . . . . . 6 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((card‘𝐴) = 𝐴𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
20 fveq2 6906 . . . . . . . . 9 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘∅))
21 aleph0 10103 . . . . . . . . 9 (ℵ‘∅) = ω
2220, 21eqtrdi 2790 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ω)
2322sseq1d 4026 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴 ↔ ω ⊆ 𝐴))
2423biimprd 248 . . . . . 6 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ω ⊆ 𝐴 → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
2519, 24anim12d 609 . . . . 5 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) → (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴)))
26 eqss 4010 . . . . 5 (𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
2725, 26imbitrrdi 252 . . . 4 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
2827com12 32 . . 3 (((card‘𝐴) = 𝐴 ∧ ω ⊆ 𝐴) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
2928ancoms 458 . 2 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
30 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
3130sseq2d 4027 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
3231onnminsb 7818 . . . . . . . . 9 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
33 vex 3481 . . . . . . . . . . 11 𝑦 ∈ V
3433sucid 6467 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
35 eleq2 2827 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ 𝑦 ∈ suc 𝑦))
3634, 35mpbiri 258 . . . . . . . . 9 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
3732, 36impel 505 . . . . . . . 8 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
3837adantl 481 . . . . . . 7 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
39 fveq2 6906 . . . . . . . . . . 11 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘suc 𝑦))
40 alephsuc 10105 . . . . . . . . . . 11 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
4139, 40sylan9eqr 2796 . . . . . . . . . 10 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (har‘(ℵ‘𝑦)))
4241eleq2d 2824 . . . . . . . . 9 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 ∈ (har‘(ℵ‘𝑦))))
4342biimpd 229 . . . . . . . 8 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ∈ (har‘(ℵ‘𝑦))))
44 elharval 9598 . . . . . . . . . 10 (𝐴 ∈ (har‘(ℵ‘𝑦)) ↔ (𝐴 ∈ On ∧ 𝐴 ≼ (ℵ‘𝑦)))
4544simprbi 496 . . . . . . . . 9 (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ≼ (ℵ‘𝑦))
46 onenon 9986 . . . . . . . . . . . 12 (𝐴 ∈ On → 𝐴 ∈ dom card)
473, 46syl 17 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 ∈ dom card)
48 alephon 10106 . . . . . . . . . . . 12 (ℵ‘𝑦) ∈ On
49 onenon 9986 . . . . . . . . . . . 12 ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
5048, 49ax-mp 5 . . . . . . . . . . 11 (ℵ‘𝑦) ∈ dom card
51 carddom2 10014 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
5247, 50, 51sylancl 586 . . . . . . . . . 10 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
53 sseq1 4020 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (card‘(ℵ‘𝑦))))
54 alephcard 10107 . . . . . . . . . . . 12 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
5554sseq2i 4024 . . . . . . . . . . 11 (𝐴 ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦))
5653, 55bitrdi 287 . . . . . . . . . 10 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
5752, 56bitr3d 281 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → (𝐴 ≼ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
5845, 57imbitrid 244 . . . . . . . 8 ((card‘𝐴) = 𝐴 → (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ⊆ (ℵ‘𝑦)))
5943, 58sylan9r 508 . . . . . . 7 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ⊆ (ℵ‘𝑦)))
6038, 59mtod 198 . . . . . 6 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
6160rexlimdvaa 3153 . . . . 5 ((card‘𝐴) = 𝐴 → (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
62 onintrab2 7816 . . . . . . . . . . . . . 14 (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
638, 62sylib 218 . . . . . . . . . . . . 13 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
64 onelon 6410 . . . . . . . . . . . . 13 (( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
6563, 64sylan 580 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
6632adantld 490 . . . . . . . . . . . 12 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
6765, 66mpcom 38 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
6848onelssi 6500 . . . . . . . . . . 11 (𝐴 ∈ (ℵ‘𝑦) → 𝐴 ⊆ (ℵ‘𝑦))
6967, 68nsyl 140 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘𝑦))
7069nrexdv 3146 . . . . . . . . 9 (𝐴 ∈ On → ¬ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
7170adantr 480 . . . . . . . 8 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
72 alephlim 10104 . . . . . . . . . . 11 (( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦))
7363, 72sylan 580 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦))
7473eleq2d 2824 . . . . . . . . 9 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦)))
75 eliun 4999 . . . . . . . . 9 (𝐴 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦) ↔ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
7674, 75bitrdi 287 . . . . . . . 8 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦)))
7771, 76mtbird 325 . . . . . . 7 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
7877ex 412 . . . . . 6 (𝐴 ∈ On → (Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
793, 78syl 17 . . . . 5 ((card‘𝐴) = 𝐴 → (Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
8061, 79jaod 859 . . . 4 ((card‘𝐴) = 𝐴 → ((∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
818, 17syl 17 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
82 alephon 10106 . . . . . . 7 (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On
83 onsseleq 6426 . . . . . . 7 ((𝐴 ∈ On ∧ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On) → (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
8482, 83mpan2 691 . . . . . 6 (𝐴 ∈ On → (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
8581, 84mpbid 232 . . . . 5 (𝐴 ∈ On → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
8685ord 864 . . . 4 (𝐴 ∈ On → (¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
873, 80, 86sylsyld 61 . . 3 ((card‘𝐴) = 𝐴 → ((∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
8887adantl 481 . 2 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ((∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
89 eloni 6395 . . . . 5 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
90 ordzsl 7865 . . . . . 6 (Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ ∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
91 3orass 1089 . . . . . 6 (( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ ∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
9290, 91bitri 275 . . . . 5 (Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
9389, 92sylib 218 . . . 4 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
943, 63, 933syl 18 . . 3 ((card‘𝐴) = 𝐴 → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
9594adantl 481 . 2 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ ∨ (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 ∨ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
9629, 88, 95mpjaod 860 1 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085   = wceq 1536  wcel 2105  wrex 3067  {crab 3432  wss 3962  c0 4338   cint 4950   ciun 4995   class class class wbr 5147  dom cdm 5688  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387  cfv 6562  ωcom 7886  cdom 8981  harchar 9593  cardccrd 9972  cale 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-har 9594  df-card 9976  df-aleph 9977
This theorem is referenced by:  cardalephex  10127  tskcard  10818  minregex  43523
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