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Theorem alephval2 10615
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 10117 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
21ralrimiv 3135 . . . 4 (𝐴 ∈ On → ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3 alephon 10112 . . . 4 (ℵ‘𝐴) ∈ On
42, 3jctil 518 . . 3 (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5 breq2 5157 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
65ralbidv 3168 . . . 4 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
76elrab 3681 . . 3 ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
84, 7sylibr 233 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
9 cardsdomelir 10016 . . . . 5 (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
10 alephcard 10113 . . . . . 6 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
1110eqcomi 2735 . . . . 5 (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
129, 11eleq2s 2844 . . . 4 (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
13 omex 9686 . . . . . 6 ω ∈ V
14 vex 3466 . . . . . 6 𝑧 ∈ V
15 entri3 10602 . . . . . 6 ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧𝑧 ≼ ω))
1613, 14, 15mp2an 690 . . . . 5 (ω ≼ 𝑧𝑧 ≼ ω)
17 carddom 10597 . . . . . . . . . 10 ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
1813, 14, 17mp2an 690 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
19 cardom 10029 . . . . . . . . . 10 (card‘ω) = ω
2019sseq1i 4008 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
2118, 20bitr3i 276 . . . . . . . 8 (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
22 cardidm 10002 . . . . . . . . . 10 (card‘(card‘𝑧)) = (card‘𝑧)
23 cardalephex 10133 . . . . . . . . . 10 (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
2422, 23mpbii 232 . . . . . . . . 9 (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
25 alephord 10118 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2625ancoms 457 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
27 breq1 5156 . . . . . . . . . . . . 13 ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2814cardid 10590 . . . . . . . . . . . . . 14 (card‘𝑧) ≈ 𝑧
29 sdomen1 9159 . . . . . . . . . . . . . 14 ((card‘𝑧) ≈ 𝑧 → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3028, 29ax-mp 5 . . . . . . . . . . . . 13 ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴))
3127, 30bitr3di 285 . . . . . . . . . . . 12 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3226, 31sylan9bb 508 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴𝑧 ≺ (ℵ‘𝐴)))
33 fveq2 6901 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
3433breq1d 5163 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
3534rspcv 3604 . . . . . . . . . . . . . 14 (𝑥𝐴 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
36 sdomirr 9152 . . . . . . . . . . . . . . 15 ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
37 sdomen2 9160 . . . . . . . . . . . . . . . . 17 ((card‘𝑧) ≈ 𝑧 → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧))
3828, 37ax-mp 5 . . . . . . . . . . . . . . . 16 ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧)
39 breq2 5157 . . . . . . . . . . . . . . . 16 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4038, 39bitr3id 284 . . . . . . . . . . . . . . 15 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4136, 40mtbiri 326 . . . . . . . . . . . . . 14 ((card‘𝑧) = (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ 𝑧)
4235, 41nsyli 157 . . . . . . . . . . . . 13 (𝑥𝐴 → ((card‘𝑧) = (ℵ‘𝑥) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4342com12 32 . . . . . . . . . . . 12 ((card‘𝑧) = (ℵ‘𝑥) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4443adantl 480 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4532, 44sylbird 259 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4645rexlimdva2 3147 . . . . . . . . 9 (𝐴 ∈ On → (∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
4724, 46syl5 34 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ (card‘𝑧) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
4821, 47biimtrid 241 . . . . . . 7 (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
4948adantr 479 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
50 ne0i 4337 . . . . . . . . . . . 12 (∅ ∈ 𝐴𝐴 ≠ ∅)
51 onelon 6401 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
52 alephgeom 10125 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
53 alephon 10112 . . . . . . . . . . . . . . . . . . 19 (ℵ‘𝑦) ∈ On
54 ssdomg 9031 . . . . . . . . . . . . . . . . . . 19 ((ℵ‘𝑦) ∈ On → (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦)))
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . . 18 (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦))
5652, 55sylbi 216 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
57 domtr 9038 . . . . . . . . . . . . . . . . 17 ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
5856, 57sylan2 591 . . . . . . . . . . . . . . . 16 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
59 domnsym 9137 . . . . . . . . . . . . . . . 16 (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6058, 59syl 17 . . . . . . . . . . . . . . 15 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6151, 60sylan2 591 . . . . . . . . . . . . . 14 ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6261expr 455 . . . . . . . . . . . . 13 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
6362ralrimiv 3135 . . . . . . . . . . . 12 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
64 r19.2z 4499 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
6564ex 411 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → (∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
6650, 63, 65syl2im 40 . . . . . . . . . . 11 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧))
67 rexnal 3090 . . . . . . . . . . 11 (∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
6866, 67imbitrdi 250 . . . . . . . . . 10 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
6968com12 32 . . . . . . . . 9 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7069expimpd 452 . . . . . . . 8 (𝑧 ≼ ω → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7170a1d 25 . . . . . . 7 (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7271com3r 87 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≼ ω → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7349, 72jaod 857 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ((ω ≼ 𝑧𝑧 ≼ ω) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
7416, 73mpi 20 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
75 breq2 5157 . . . . . . . 8 (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
7675ralbidv 3168 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7776elrab 3681 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7877simprbi 495 . . . . 5 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} → ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
7978con3i 154 . . . 4 (¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
8012, 74, 79syl56 36 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥}))
8180ralrimiv 3135 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
82 ssrab2 4076 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
83 oneqmini 6428 . . 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}))
8482, 83ax-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})
858, 81, 84syl2an2r 683 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 205  wa 394  wo 845   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  wss 3947  c0 4325   cint 4954   class class class wbr 5153  Oncon0 6376  cfv 6554  ωcom 7876  cen 8971  cdom 8972  csdm 8973  cardccrd 9978  cale 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-ac2 10506
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9553  df-har 9600  df-card 9982  df-aleph 9983  df-ac 10159
This theorem is referenced by: (None)
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