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Theorem cardaleph 10129
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 9984 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2829 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 233 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 alephle 10128 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
5 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
65sseq2d 4016 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
76rspcev 3622 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐴 ⊆ (ℵ‘𝐴)) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
84, 7mpdan 687 . . . . . . . 8 (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
9 nfcv 2905 . . . . . . . . . 10 𝑥𝐴
10 nfcv 2905 . . . . . . . . . . 11 𝑥
11 nfrab1 3457 . . . . . . . . . . . 12 𝑥{𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1211nfint 4956 . . . . . . . . . . 11 𝑥 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1310, 12nffv 6916 . . . . . . . . . 10 𝑥(ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
149, 13nfss 3976 . . . . . . . . 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 4016 . . . . . . . . 9 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
1714, 16onminsb 7814 . . . . . . . 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 10106 . . . . . . . . 9 (ℵ‘∅) = ω
2220, 21eqtrdi 2793 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ω)
2322sseq1d 4015 . . . . . . 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 3999 . . . . 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 4016 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
3231onnminsb 7819 . . . . . . . . 9 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
33 vex 3484 . . . . . . . . . . 11 𝑦 ∈ V
3433sucid 6466 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
35 eleq2 2830 . . . . . . . . . 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 10108 . . . . . . . . . . 11 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
4139, 40sylan9eqr 2799 . . . . . . . . . 10 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (har‘(ℵ‘𝑦)))
4241eleq2d 2827 . . . . . . . . 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 9601 . . . . . . . . . 10 (𝐴 ∈ (har‘(ℵ‘𝑦)) ↔ (𝐴 ∈ On ∧ 𝐴 ≼ (ℵ‘𝑦)))
4544simprbi 496 . . . . . . . . 9 (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ≼ (ℵ‘𝑦))
46 onenon 9989 . . . . . . . . . . . 12 (𝐴 ∈ On → 𝐴 ∈ dom card)
473, 46syl 17 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 ∈ dom card)
48 alephon 10109 . . . . . . . . . . . 12 (ℵ‘𝑦) ∈ On
49 onenon 9989 . . . . . . . . . . . 12 ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
5048, 49ax-mp 5 . . . . . . . . . . 11 (ℵ‘𝑦) ∈ dom card
51 carddom2 10017 . . . . . . . . . . 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 4009 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (card‘(ℵ‘𝑦))))
54 alephcard 10110 . . . . . . . . . . . 12 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
5554sseq2i 4013 . . . . . . . . . . 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 3156 . . . . 5 ((card‘𝐴) = 𝐴 → (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
62 onintrab2 7817 . . . . . . . . . . . . . 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 6409 . . . . . . . . . . . . 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 6499 . . . . . . . . . . 11 (𝐴 ∈ (ℵ‘𝑦) → 𝐴 ⊆ (ℵ‘𝑦))
6967, 68nsyl 140 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘𝑦))
7069nrexdv 3149 . . . . . . . . 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 10107 . . . . . . . . . . 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 2827 . . . . . . . . 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 4995 . . . . . . . . 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 860 . . . 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 10109 . . . . . . 7 (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On
83 onsseleq 6425 . . . . . . 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 865 . . . 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 6394 . . . . 5 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
90 ordzsl 7866 . . . . . 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 1090 . . . . . 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 861 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 848  w3o 1086   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  wss 3951  c0 4333   cint 4946   ciun 4991   class class class wbr 5143  dom cdm 5685  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  cfv 6561  ωcom 7887  cdom 8983  harchar 9596  cardccrd 9975  cale 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980
This theorem is referenced by:  cardalephex  10130  tskcard  10821  minregex  43547
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