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Theorem cardaleph 9189
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 9047 . . . . . . . . 9 (card‘𝐴) ∈ On
2 eleq1 2869 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On))
31, 2mpbii 224 . . . . . . . 8 ((card‘𝐴) = 𝐴𝐴 ∈ On)
4 alephle 9188 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
5 fveq2 6402 . . . . . . . . . . 11 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
65sseq2d 3824 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
76rspcev 3498 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐴 ⊆ (ℵ‘𝐴)) → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
84, 7mpdan 670 . . . . . . . 8 (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥))
9 nfcv 2944 . . . . . . . . . 10 𝑥𝐴
10 nfcv 2944 . . . . . . . . . . 11 𝑥
11 nfrab1 3307 . . . . . . . . . . . 12 𝑥{𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1211nfint 4672 . . . . . . . . . . 11 𝑥 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}
1310, 12nffv 6412 . . . . . . . . . 10 𝑥(ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
149, 13nfss 3785 . . . . . . . . 9 𝑥 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
15 fveq2 6402 . . . . . . . . . 10 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (ℵ‘𝑥) = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
1615sseq2d 3824 . . . . . . . . 9 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
1714, 16onminsb 7223 . . . . . . . 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 6402 . . . . . . . . 9 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘∅))
21 aleph0 9166 . . . . . . . . 9 (ℵ‘∅) = ω
2220, 21syl6eq 2852 . . . . . . . 8 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = ω)
2322sseq1d 3823 . . . . . . 7 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → ((ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴 ↔ ω ⊆ 𝐴))
2423biimprd 239 . . . . . 6 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → (ω ⊆ 𝐴 → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
2519, 24anim12d 598 . . . . 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 3807 . . . . 5 (𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ (𝐴 ⊆ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∧ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ⊆ 𝐴))
2725, 26syl6ibr 243 . . . 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 448 . 2 ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = ∅ → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
30 vex 3390 . . . . . . . . . . . 12 𝑦 ∈ V
3130sucid 6011 . . . . . . . . . . 11 𝑦 ∈ suc 𝑦
32 eleq2 2870 . . . . . . . . . . 11 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ↔ 𝑦 ∈ suc 𝑦))
3331, 32mpbiri 249 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
34 fveq2 6402 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
3534sseq2d 3824 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐴 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
3635onnminsb 7228 . . . . . . . . . 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 395 . . . . . . . 8 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
3938adantl 469 . . . . . . 7 (((card‘𝐴) = 𝐴 ∧ (𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦)) → ¬ 𝐴 ⊆ (ℵ‘𝑦))
40 fveq2 6402 . . . . . . . . . . 11 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (ℵ‘suc 𝑦))
41 alephsuc 9168 . . . . . . . . . . 11 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
4240, 41sylan9eqr 2858 . . . . . . . . . 10 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = (har‘(ℵ‘𝑦)))
4342eleq2d 2867 . . . . . . . . 9 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ↔ 𝐴 ∈ (har‘(ℵ‘𝑦))))
4443biimpd 220 . . . . . . . 8 ((𝑦 ∈ On ∧ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦) → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝐴 ∈ (har‘(ℵ‘𝑦))))
45 elharval 8701 . . . . . . . . . 10 (𝐴 ∈ (har‘(ℵ‘𝑦)) ↔ (𝐴 ∈ On ∧ 𝐴 ≼ (ℵ‘𝑦)))
4645simprbi 486 . . . . . . . . 9 (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ≼ (ℵ‘𝑦))
47 onenon 9052 . . . . . . . . . . . 12 (𝐴 ∈ On → 𝐴 ∈ dom card)
483, 47syl 17 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 ∈ dom card)
49 alephon 9169 . . . . . . . . . . . 12 (ℵ‘𝑦) ∈ On
50 onenon 9052 . . . . . . . . . . . 12 ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
5149, 50ax-mp 5 . . . . . . . . . . 11 (ℵ‘𝑦) ∈ dom card
52 carddom2 9080 . . . . . . . . . . 11 ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
5348, 51, 52sylancl 576 . . . . . . . . . 10 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
54 sseq1 3817 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (card‘(ℵ‘𝑦))))
55 alephcard 9170 . . . . . . . . . . . 12 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
5655sseq2i 3821 . . . . . . . . . . 11 (𝐴 ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦))
5754, 56syl6bb 278 . . . . . . . . . 10 ((card‘𝐴) = 𝐴 → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
5853, 57bitr3d 272 . . . . . . . . 9 ((card‘𝐴) = 𝐴 → (𝐴 ≼ (ℵ‘𝑦) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
5946, 58syl5ib 235 . . . . . . . 8 ((card‘𝐴) = 𝐴 → (𝐴 ∈ (har‘(ℵ‘𝑦)) → 𝐴 ⊆ (ℵ‘𝑦)))
6044, 59sylan9r 500 . . . . . . 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 3216 . . . . 5 ((card‘𝐴) = 𝐴 → (∃𝑦 ∈ On {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} = suc 𝑦 → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
63 onintrab2 7226 . . . . . . . . . . . . . 14 (∃𝑥 ∈ On 𝐴 ⊆ (ℵ‘𝑥) ↔ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
648, 63sylib 209 . . . . . . . . . . . . 13 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On)
65 onelon 5955 . . . . . . . . . . . . 13 (( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
6664, 65sylan 571 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → 𝑦 ∈ On)
6736adantld 480 . . . . . . . . . . . 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 6043 . . . . . . . . . . 11 (𝐴 ∈ (ℵ‘𝑦) → 𝐴 ⊆ (ℵ‘𝑦))
7068, 69nsyl 137 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘𝑦))
7170nrexdv 3184 . . . . . . . . 9 (𝐴 ∈ On → ¬ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
7271adantr 468 . . . . . . . 8 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
73 alephlim 9167 . . . . . . . . . . 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 571 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) = 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦))
7574eleq2d 2867 . . . . . . . . 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 4709 . . . . . . . . 9 (𝐴 𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} (ℵ‘𝑦) ↔ ∃𝑦 {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}𝐴 ∈ (ℵ‘𝑦))
7775, 76syl6bb 278 . . . . . . . 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 316 . . . . . . 7 ((𝐴 ∈ On ∧ Lim {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) → ¬ 𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
7978ex 399 . . . . . 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 877 . . . 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 9169 . . . . . . 7 (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∈ On
84 onsseleq 5972 . . . . . . 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 674 . . . . . 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 223 . . . . 5 (𝐴 ∈ On → (𝐴 ∈ (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}) ∨ 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})))
8786ord 882 . . . 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 469 . 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 5940 . . . . 5 ( {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)} ∈ On → Ord {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)})
91 ordzsl 7269 . . . . . 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 1103 . . . . . 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 266 . . . . 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 209 . . . 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 469 . 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 878 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 197  wa 384  wo 865  w3o 1099   = wceq 1637  wcel 2155  wrex 3093  {crab 3096  wss 3763  c0 4110   cint 4662   ciun 4705   class class class wbr 4837  dom cdm 5305  Ord word 5929  Oncon0 5930  Lim wlim 5931  suc csuc 5932  cfv 6095  ωcom 7289  cdom 8184  harchar 8694  cardccrd 9038  cale 9039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-inf2 8779
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-isom 6104  df-riota 6829  df-om 7290  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-er 7973  df-en 8187  df-dom 8188  df-sdom 8189  df-fin 8190  df-oi 8648  df-har 8696  df-card 9042  df-aleph 9043
This theorem is referenced by:  cardalephex  9190  tskcard  9882
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