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Theorem cardaleph 8950
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 8808 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2718 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 223 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 alephle 8949 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
5 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
65sseq2d 3666 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
76rspcev 3340 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐴 ⊆ (ℵ‘𝐴)) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
84, 7mpdan 703 . . . . . . . 8 (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
9 nfcv 2793 . . . . . . . . . 10 𝑥𝐴
10 nfcv 2793 . . . . . . . . . . 11 𝑥
11 nfrab1 3152 . . . . . . . . . . . 12 𝑥{𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1211nfint 4518 . . . . . . . . . . 11 𝑥 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1310, 12nffv 6236 . . . . . . . . . 10 𝑥(ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
149, 13nfss 3629 . . . . . . . . 9 𝑥 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
15 fveq2 6229 . . . . . . . . . 10 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (ℵ‘𝑥) = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
1615sseq2d 3666 . . . . . . . . 9 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
1714, 16onminsb 7041 . . . . . . . 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 6229 . . . . . . . . 9 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘∅))
21 aleph0 8927 . . . . . . . . 9 (ℵ‘∅) = ω
2220, 21syl6eq 2701 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ω)
2322sseq1d 3665 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴 ↔ ω ⊆ 𝐴))
2423biimprd 238 . . . . . 6 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ω ⊆ 𝐴 → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
2519, 24anim12d 585 . . . . 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 3651 . . . . 5 (𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
2725, 26syl6ibr 242 . . . 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 468 . 2 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
30 vex 3234 . . . . . . . . . . . 12 𝑦 ∈ V
3130sucid 5842 . . . . . . . . . . 11 𝑦 ∈ suc 𝑦
32 eleq2 2719 . . . . . . . . . . 11 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ 𝑦 ∈ suc 𝑦))
3331, 32mpbiri 248 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
34 fveq2 6229 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
3534sseq2d 3666 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
3635onnminsb 7046 . . . . . . . . . 10 (𝑦 ∈ On → (𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
3733, 36syl5 34 . . . . . . . . 9 (𝑦 ∈ On → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
3837imp 444 . . . . . . . 8 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
3938adantl 481 . . . . . . 7 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
40 fveq2 6229 . . . . . . . . . . 11 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘suc 𝑦))
41 alephsuc 8929 . . . . . . . . . . 11 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
4240, 41sylan9eqr 2707 . . . . . . . . . 10 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (har‘(ℵ‘𝑦)))
4342eleq2d 2716 . . . . . . . . 9 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 ∈ (har‘(ℵ‘𝑦))))
4443biimpd 219 . . . . . . . 8 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ∈ (har‘(ℵ‘𝑦))))
45 elharval 8509 . . . . . . . . . 10 (𝐴 ∈ (har‘(ℵ‘𝑦)) ↔ (𝐴 ∈ On ∧ 𝐴 ≼ (ℵ‘𝑦)))
4645simprbi 479 . . . . . . . . 9 (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ≼ (ℵ‘𝑦))
47 onenon 8813 . . . . . . . . . . . 12 (𝐴 ∈ On → 𝐴 ∈ dom card)
483, 47syl 17 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 ∈ dom card)
49 alephon 8930 . . . . . . . . . . . 12 (ℵ‘𝑦) ∈ On
50 onenon 8813 . . . . . . . . . . . 12 ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
5149, 50ax-mp 5 . . . . . . . . . . 11 (ℵ‘𝑦) ∈ dom card
52 carddom2 8841 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
5348, 51, 52sylancl 695 . . . . . . . . . 10 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
54 sseq1 3659 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (card‘(ℵ‘𝑦))))
55 alephcard 8931 . . . . . . . . . . . 12 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
5655sseq2i 3663 . . . . . . . . . . 11 (𝐴 ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦))
5754, 56syl6bb 276 . . . . . . . . . 10 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
5853, 57bitr3d 270 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → (𝐴 ≼ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
5946, 58syl5ib 234 . . . . . . . 8 ((card‘𝐴) = 𝐴 → (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ⊆ (ℵ‘𝑦)))
6044, 59sylan9r 691 . . . . . . 7 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ⊆ (ℵ‘𝑦)))
6139, 60mtod 189 . . . . . 6 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
6261rexlimdvaa 3061 . . . . 5 ((card‘𝐴) = 𝐴 → (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
63 onintrab2 7044 . . . . . . . . . . . . . 14 (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
648, 63sylib 208 . . . . . . . . . . . . 13 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
65 onelon 5786 . . . . . . . . . . . . 13 (( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
6664, 65sylan 487 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
6736adantld 482 . . . . . . . . . . . 12 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ⊆ (ℵ‘𝑦)))
6866, 67mpcom 38 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
6949onelssi 5874 . . . . . . . . . . 11 (𝐴 ∈ (ℵ‘𝑦) → 𝐴 ⊆ (ℵ‘𝑦))
7068, 69nsyl 135 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘𝑦))
7170nrexdv 3030 . . . . . . . . 9 (𝐴 ∈ On → ¬ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
7271adantr 480 . . . . . . . 8 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
73 alephlim 8928 . . . . . . . . . . 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))
7464, 73sylan 487 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦))
7574eleq2d 2716 . . . . . . . . 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)))
76 eliun 4556 . . . . . . . . 9 (𝐴 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦) ↔ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
7775, 76syl6bb 276 . . . . . . . 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)))
7872, 77mtbird 314 . . . . . . 7 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
7978ex 449 . . . . . 6 (𝐴 ∈ On → (Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
803, 79syl 17 . . . . 5 ((card‘𝐴) = 𝐴 → (Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
8162, 80jaod 394 . . . 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)})))
828, 17syl 17 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
83 alephon 8930 . . . . . . 7 (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On
84 onsseleq 5803 . . . . . . 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)}))))
8583, 84mpan2 707 . . . . . 6 (𝐴 ∈ On → (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))))
8682, 85mpbid 222 . . . . 5 (𝐴 ∈ On → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
8786ord 391 . . . 4 (𝐴 ∈ On → (¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
883, 81, 87sylsyld 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)})))
8988adantl 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)})))
90 eloni 5771 . . . . 5 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
91 ordzsl 7087 . . . . . 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)}))
92 3orass 1057 . . . . . 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)})))
9391, 92bitri 264 . . . . 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)})))
9490, 93sylib 208 . . . 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)})))
953, 64, 943syl 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)})))
9695adantl 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)})))
9729, 89, 96mpjaod 395 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 196  wo 382  wa 383  w3o 1053   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  wss 3607  c0 3948   cint 4507   ciun 4552   class class class wbr 4685  dom cdm 5143  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  cfv 5926  ωcom 7107  cdom 7995  harchar 8502  cardccrd 8799  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-card 8803  df-aleph 8804
This theorem is referenced by:  cardalephex  8951  tskcard  9641
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