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Theorem alephval2 9983
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 9489 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
21ralrimiv 3181 . . . 4 (𝐴 ∈ On → ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴))
3 alephon 9484 . . . 4 (ℵ‘𝐴) ∈ On
42, 3jctil 520 . . 3 (𝐴 ∈ On → ((ℵ‘𝐴) ∈ On ∧ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
5 breq2 5062 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
65ralbidv 3197 . . . 4 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ (ℵ‘𝐴)))
76elrab 3679 . . 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 9391 . . . . 5 (𝑧 ∈ (card‘(ℵ‘𝐴)) → 𝑧 ≺ (ℵ‘𝐴))
10 alephcard 9485 . . . . . 6 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
1110eqcomi 2830 . . . . 5 (ℵ‘𝐴) = (card‘(ℵ‘𝐴))
129, 11eleq2s 2931 . . . 4 (𝑧 ∈ (ℵ‘𝐴) → 𝑧 ≺ (ℵ‘𝐴))
13 omex 9095 . . . . . 6 ω ∈ V
14 vex 3498 . . . . . 6 𝑧 ∈ V
15 entri3 9970 . . . . . 6 ((ω ∈ V ∧ 𝑧 ∈ V) → (ω ≼ 𝑧𝑧 ≼ ω))
1613, 14, 15mp2an 688 . . . . 5 (ω ≼ 𝑧𝑧 ≼ ω)
17 carddom 9965 . . . . . . . . . 10 ((ω ∈ V ∧ 𝑧 ∈ V) → ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧))
1813, 14, 17mp2an 688 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ≼ 𝑧)
19 cardom 9404 . . . . . . . . . 10 (card‘ω) = ω
2019sseq1i 3994 . . . . . . . . 9 ((card‘ω) ⊆ (card‘𝑧) ↔ ω ⊆ (card‘𝑧))
2118, 20bitr3i 278 . . . . . . . 8 (ω ≼ 𝑧 ↔ ω ⊆ (card‘𝑧))
22 cardidm 9377 . . . . . . . . . 10 (card‘(card‘𝑧)) = (card‘𝑧)
23 cardalephex 9505 . . . . . . . . . 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 9490 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2625ancoms 459 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑥𝐴 ↔ (ℵ‘𝑥) ≺ (ℵ‘𝐴)))
2714cardid 9958 . . . . . . . . . . . . . 14 (card‘𝑧) ≈ 𝑧
28 sdomen1 8650 . . . . . . . . . . . . . 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 5061 . . . . . . . . . . . . 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 6664 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (ℵ‘𝑦) = (ℵ‘𝑥))
3433breq1d 5068 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((ℵ‘𝑦) ≺ 𝑧 ↔ (ℵ‘𝑥) ≺ 𝑧))
3534rspcv 3617 . . . . . . . . . . . . . 14 (𝑥𝐴 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧 → (ℵ‘𝑥) ≺ 𝑧))
36 sdomirr 8643 . . . . . . . . . . . . . . 15 ¬ (ℵ‘𝑥) ≺ (ℵ‘𝑥)
37 sdomen2 8651 . . . . . . . . . . . . . . . . 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 5062 . . . . . . . . . . . . . . . 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 3287 . . . . . . . . 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 4299 . . . . . . . . . . . 12 (∅ ∈ 𝐴𝐴 ≠ ∅)
51 onelon 6210 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
52 alephgeom 9497 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On ↔ ω ⊆ (ℵ‘𝑦))
53 alephon 9484 . . . . . . . . . . . . . . . . . . 19 (ℵ‘𝑦) ∈ On
54 ssdomg 8544 . . . . . . . . . . . . . . . . . . 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 8551 . . . . . . . . . . . . . . . . 17 ((𝑧 ≼ ω ∧ ω ≼ (ℵ‘𝑦)) → 𝑧 ≼ (ℵ‘𝑦))
5856, 57sylan2 592 . . . . . . . . . . . . . . . 16 ((𝑧 ≼ ω ∧ 𝑦 ∈ On) → 𝑧 ≼ (ℵ‘𝑦))
59 domnsym 8632 . . . . . . . . . . . . . . . 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 3181 . . . . . . . . . . . 12 ((𝑧 ≼ ω ∧ 𝐴 ∈ On) → ∀𝑦𝐴 ¬ (ℵ‘𝑦) ≺ 𝑧)
64 r19.2z 4438 . . . . . . . . . . . . 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 3238 . . . . . . . . . . 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 5062 . . . . . . . 8 (𝑥 = 𝑧 → ((ℵ‘𝑦) ≺ 𝑥 ↔ (ℵ‘𝑦) ≺ 𝑧))
7675ralbidv 3197 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥 ↔ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑧))
7776elrab 3679 . . . . . 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 3181 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
82 ssrab2 4055 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥} ⊆ On
83 oneqmini 6236 . . 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 3016  wral 3138  wrex 3139  {crab 3142  Vcvv 3495  wss 3935  c0 4290   cint 4869   class class class wbr 5058  Oncon0 6185  cfv 6349  ωcom 7568  cen 8495  cdom 8496  csdm 8497  cardccrd 9353  cale 9354
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-inf2 9093  ax-ac2 9874
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-oi 8963  df-har 9011  df-card 9357  df-aleph 9358  df-ac 9531
This theorem is referenced by: (None)
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