MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephval2 Structured version   Visualization version   GIF version

Theorem alephval2 9982
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 9488 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
21ralrimiv 3178 . . . 4 (𝐴 ∈ On → ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3 alephon 9483 . . . 4 (ℵ‘𝐴) ∈ On
42, 3jctil 520 . . 3 (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5 breq2 5061 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
65ralbidv 3194 . . . 4 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
76elrab 3677 . . 3 ((ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
84, 7sylibr 235 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
9 cardsdomelir 9390 . . . . 5 (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
10 alephcard 9484 . . . . . 6 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
1110eqcomi 2827 . . . . 5 (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
129, 11eleq2s 2928 . . . 4 (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
13 omex 9094 . . . . . 6 ω ∈ V
14 vex 3495 . . . . . 6 𝑧 ∈ V
15 entri3 9969 . . . . . 6 ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧𝑧 ≼ ω))
1613, 14, 15mp2an 688 . . . . 5 (ω ≼ 𝑧𝑧 ≼ ω)
17 carddom 9964 . . . . . . . . . 10 ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
1813, 14, 17mp2an 688 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
19 cardom 9403 . . . . . . . . . 10 (card‘ω) = ω
2019sseq1i 3992 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
2118, 20bitr3i 278 . . . . . . . 8 (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
22 cardidm 9376 . . . . . . . . . 10 (card‘(card‘𝑧)) = (card‘𝑧)
23 cardalephex 9504 . . . . . . . . . 10 (ω ⊆ (card‘𝑧) → ((card‘(card‘𝑧)) = (card‘𝑧) ↔ ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥)))
2422, 23mpbii 234 . . . . . . . . 9 (ω ⊆ (card‘𝑧) → ∃𝑥 ∈ On (card‘𝑧) = (ℵ‘𝑥))
25 alephord 9489 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2625ancoms 459 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2714cardid 9957 . . . . . . . . . . . . . 14 (card‘𝑧) ≈ 𝑧
28 sdomen1 8649 . . . . . . . . . . . . . 14 ((card‘𝑧) ≈ 𝑧 → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
2927, 28ax-mp 5 . . . . . . . . . . . . 13 ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴))
30 breq1 5060 . . . . . . . . . . . . 13 ((card‘𝑧) = (ℵ‘𝑥) → ((card‘𝑧) ≺ (ℵ‘𝐴) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
3129, 30syl5rbbr 287 . . . . . . . . . . . 12 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (ℵ‘𝐴) ↔ 𝑧 ≺ (ℵ‘𝐴)))
3226, 31sylan9bb 510 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴𝑧 ≺ (ℵ‘𝐴)))
33 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
3433breq1d 5067 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
3534rspcv 3615 . . . . . . . . . . . . . 14 (𝑥𝐴 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
36 sdomirr 8642 . . . . . . . . . . . . . . 15 ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
37 sdomen2 8650 . . . . . . . . . . . . . . . . 17 ((card‘𝑧) ≈ 𝑧 → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧))
3827, 37ax-mp 5 . . . . . . . . . . . . . . . 16 ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ 𝑧)
39 breq2 5061 . . . . . . . . . . . . . . . 16 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ (card‘𝑧) ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4038, 39syl5bbr 286 . . . . . . . . . . . . . . 15 ((card‘𝑧) = (ℵ‘𝑥) → ((ℵ‘𝑥) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝑥)))
4136, 40mtbiri 328 . . . . . . . . . . . . . 14 ((card‘𝑧) = (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ≺ 𝑧)
4235, 41nsyli 160 . . . . . . . . . . . . 13 (𝑥𝐴 → ((card‘𝑧) = (ℵ‘𝑥) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4342com12 32 . . . . . . . . . . . 12 ((card‘𝑧) = (ℵ‘𝑥) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4443adantl 482 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑥𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4532, 44sylbird 261 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (card‘𝑧) = (ℵ‘𝑥)) → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
4645rexlimdva2 3284 . . . . . . . . 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, 47syl5bi 243 . . . . . . 7 (𝐴 ∈ On → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
4948adantr 481 . . . . . 6 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ω ≼ 𝑧 → (𝑧 ≺ (ℵ‘𝐴) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)))
50 ne0i 4297 . . . . . . . . . . . 12 (∅ ∈ 𝐴𝐴 ≠ ∅)
51 onelon 6209 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
52 alephgeom 9496 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
53 alephon 9483 . . . . . . . . . . . . . . . . . . 19 (ℵ‘𝑦) ∈ On
54 ssdomg 8543 . . . . . . . . . . . . . . . . . . 19 ((ℵ‘𝑦) ∈ On → (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦)))
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . . 18 (ω ⊆ (ℵ‘𝑦) → ω ≼ (ℵ‘𝑦))
5652, 55sylbi 218 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → ω ≼ (ℵ‘𝑦))
57 domtr 8550 . . . . . . . . . . . . . . . . 17 ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
5856, 57sylan2 592 . . . . . . . . . . . . . . . 16 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
59 domnsym 8631 . . . . . . . . . . . . . . . 16 (𝑧 ≼ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6058, 59syl 17 . . . . . . . . . . . . . . 15 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6151, 60sylan2 592 . . . . . . . . . . . . . 14 ((𝑧 ≼ ω ∧ (𝐴 ∈ On ∧ 𝑦𝐴)) → ¬ (ℵ‘𝑦) ≺ 𝑧)
6261expr 457 . . . . . . . . . . . . 13 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (𝑦𝐴 → ¬ (ℵ‘𝑦) ≺ 𝑧))
6362ralrimiv 3178 . . . . . . . . . . . 12 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
64 r19.2z 4436 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧) → ∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
6564ex 413 . . . . . . . . . . . 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 3235 . . . . . . . . . . 11 (∃𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧 ↔ ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
6866, 67syl6ib 252 . . . . . . . . . 10 (∅ ∈ 𝐴 → ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
6968com12 32 . . . . . . . . 9 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → ¬ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7069expimpd 454 . . . . . . . 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 853 . . . . 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 5061 . . . . . . . 8 (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
7675ralbidv 3194 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7776elrab 3677 . . . . . 6 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ↔ (𝑧 ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7877simprbi 497 . . . . 5 (𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} → ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧)
7978con3i 157 . . . 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 3178 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
82 ssrab2 4053 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
83 oneqmini 6235 . . 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 681 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 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  c0 4288   cint 4867   class class class wbr 5057  Oncon0 6184  cfv 6348  ωcom 7569  cen 8494  cdom 8495  csdm 8496  cardccrd 9352  cale 9353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-har 9010  df-card 9356  df-aleph 9357  df-ac 9530
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator