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Theorem alephval2 9273
 Description: An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.)
Assertion
Ref Expression
alephval2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem alephval2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 alephordi 8780 . . . . . 6 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
21ralrimiv 2948 . . . . 5 (𝐴 ∈ On → ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3 alephon 8775 . . . . 5 (ℵ‘𝐴) ∈ On
42, 3jctil 558 . . . 4 (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5 breq2 4587 . . . . . 6 (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
65ralbidv 2969 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
76elrab 3331 . . . 4 ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
84, 7sylibr 223 . . 3 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
98adantr 480 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
10 cardsdomelir 8682 . . . . 5 (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
11 alephcard 8776 . . . . . 6 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
1211eqcomi 2619 . . . . 5 (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
1310, 12eleq2s 2706 . . . 4 (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
14 omex 8423 . . . . . 6 ω ∈ V
15 vex 3176 . . . . . 6 𝑧 ∈ V
16 entri3 9260 . . . . . 6 ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧𝑧 ≼ ω))
1714, 15, 16mp2an 704 . . . . 5 (ω ≼ 𝑧𝑧 ≼ ω)
18 carddom 9255 . . . . . . . . . 10 ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
1914, 15, 18mp2an 704 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
20 cardom 8695 . . . . . . . . . 10 (card‘ω) = ω
2120sseq1i 3592 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
2219, 21bitr3i 265 . . . . . . . 8 (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
23 cardidm 8668 . . . . . . . . . 10 (card‘(card‘𝑧)) = (card‘𝑧)
24 cardalephex 8796 . . . . . . . . . 10 (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
2523, 24mpbii 222 . . . . . . . . 9 (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
26 alephord 8781 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2726ancoms 468 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2815cardid 9248 . . . . . . . . . . . . . . 15 (card‘𝑧) ≈ 𝑧
29 sdomen1 7989 . . . . . . . . . . . . . . 15 ((card‘𝑧) ≈ 𝑧 → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3028, 29ax-mp 5 . . . . . . . . . . . . . 14 ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴))
31 breq1 4586 . . . . . . . . . . . . . 14 ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
3230, 31syl5rbbr 274 . . . . . . . . . . . . 13 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3327, 32sylan9bb 732 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴𝑧 ≺ (ℵ‘𝐴)))
34 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
3534breq1d 4593 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
3635rspcv 3278 . . . . . . . . . . . . . . 15 (𝑥𝐴 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
37 sdomirr 7982 . . . . . . . . . . . . . . . 16 ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
38 sdomen2 7990 . . . . . . . . . . . . . . . . . 18 ((card‘𝑧) ≈ 𝑧 → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧))
3928, 38ax-mp 5 . . . . . . . . . . . . . . . . 17 ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧)
40 breq2 4587 . . . . . . . . . . . . . . . . 17 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4139, 40syl5bbr 273 . . . . . . . . . . . . . . . 16 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4237, 41mtbiri 316 . . . . . . . . . . . . . . 15 ((card‘𝑧) = (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ 𝑧)
4336, 42nsyli 154 . . . . . . . . . . . . . 14 (𝑥𝐴 → ((card‘𝑧) = (ℵ‘𝑥) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4443com12 32 . . . . . . . . . . . . 13 ((card‘𝑧) = (ℵ‘𝑥) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4544adantl 481 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4633, 45sylbird 249 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4746exp31 628 . . . . . . . . . 10 (𝐴 ∈ On → (𝑥 ∈ On → ((card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))))
4847rexlimdv 3012 . . . . . . . . 9 (𝐴 ∈ On → (∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
4925, 48syl5 33 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ (card‘𝑧) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
5022, 49syl5bi 231 . . . . . . 7 (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
5150adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
52 ne0i 3880 . . . . . . . . . . . 12 (∅ ∈ 𝐴𝐴 ≠ ∅)
53 onelon 5665 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
54 alephgeom 8788 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
55 alephon 8775 . . . . . . . . . . . . . . . . . . 19 (ℵ‘𝑦) ∈ On
56 ssdomg 7887 . . . . . . . . . . . . . . . . . . 19 ((ℵ‘𝑦) ∈ On → (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦)))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . . 18 (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦))
5854, 57sylbi 206 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
59 domtr 7895 . . . . . . . . . . . . . . . . 17 ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
6058, 59sylan2 490 . . . . . . . . . . . . . . . 16 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
61 domnsym 7971 . . . . . . . . . . . . . . . 16 (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6260, 61syl 17 . . . . . . . . . . . . . . 15 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6353, 62sylan2 490 . . . . . . . . . . . . . 14 ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6463expr 641 . . . . . . . . . . . . 13 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
6564ralrimiv 2948 . . . . . . . . . . . 12 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
66 r19.2z 4012 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
6766ex 449 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → (∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
6852, 65, 67syl2im 39 . . . . . . . . . . 11 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
69 rexnal 2978 . . . . . . . . . . 11 (∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
7068, 69syl6ib 240 . . . . . . . . . 10 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7170com12 32 . . . . . . . . 9 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7271expimpd 627 . . . . . . . 8 (𝑧 ≼ ω → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7372a1d 25 . . . . . . 7 (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7473com3r 85 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7551, 74jaod 394 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((ω ≼ 𝑧𝑧 ≼ ω) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7617, 75mpi 20 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
77 breq2 4587 . . . . . . . 8 (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
7877ralbidv 2969 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7978elrab 3331 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
8079simprbi 479 . . . . 5 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} → ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
8180con3i 149 . . . 4 (¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
8213, 76, 81syl56 35 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}))
8382ralrimiv 2948 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
84 ssrab2 3650 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
85 oneqmini 5693 . . 3 ({𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On → (((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}))
8684, 85ax-mp 5 . 2 (((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
879, 83, 86syl2anc 691 1 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   class class class wbr 4583  Oncon0 5640  ‘cfv 5804  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644  ℵcale 8645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649  df-ac 8822 This theorem is referenced by: (None)
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