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Theorem alephval3 9115
Description: An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)
Assertion
Ref Expression
alephval3 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem alephval3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 alephcard 9075 . . . 4 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3 alephgeom 9087 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
43biimpi 206 . . 3 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5 alephord2i 9082 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6 elirr 8659 . . . . . . 7 ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7 eleq2 2820 . . . . . . 7 ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
86, 7mtbiri 316 . . . . . 6 ((ℵ‘𝐴) = (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (ℵ‘𝐴))
98con2i 134 . . . . 5 ((ℵ‘𝑦) ∈ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
105, 9syl6 35 . . . 4 (𝐴 ∈ On → (𝑦𝐴 → ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1110ralrimiv 3095 . . 3 (𝐴 ∈ On → ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
12 fvex 6354 . . . 4 (ℵ‘𝐴) ∈ V
13 fveq2 6344 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14 id 22 . . . . . 6 (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
1513, 14eqeq12d 2767 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16 sseq2 3760 . . . . 5 (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17 eqeq1 2756 . . . . . . 7 (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
1817notbid 307 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1918ralbidv 3116 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
2015, 16, 193anbi123d 1540 . . . 4 (𝑥 = (ℵ‘𝐴) → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘(ℵ‘𝐴)) = (ℵ‘𝐴) ∧ ω ⊆ (ℵ‘𝐴) ∧ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))))
2112, 20elab 3482 . . 3 ((ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘(ℵ‘𝐴)) = (ℵ‘𝐴) ∧ ω ⊆ (ℵ‘𝐴) ∧ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
222, 4, 11, 21syl3anbrc 1426 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23 cardalephex 9095 . . . . . . . . . 10 (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
2423biimpac 504 . . . . . . . . 9 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
25 eleq1 2819 . . . . . . . . . . . . . . . 16 (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
26 alephord2 9081 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
2726bicomd 213 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2825, 27sylan9bbr 739 . . . . . . . . . . . . . . 15 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2928biimpcd 239 . . . . . . . . . . . . . 14 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦𝐴))
30 simpr 479 . . . . . . . . . . . . . . 15 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
3130a1i 11 . . . . . . . . . . . . . 14 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦)))
3229, 31jcad 556 . . . . . . . . . . . . 13 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3332exp4c 637 . . . . . . . . . . . 12 (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝐴 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))))
3433com3r 87 . . . . . . . . . . 11 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))))
3534imp4b 614 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3635reximdv2 3144 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦)))
3724, 36syl5 34 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦)))
3837imp 444 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦))
39 dfrex2 3126 . . . . . . 7 (∃𝑦𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
4038, 39sylib 208 . . . . . 6 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
41 nan 605 . . . . . 6 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))) ↔ (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
4240, 41mpbir 221 . . . . 5 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
4342ex 449 . . . 4 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
44 vex 3335 . . . . . . 7 𝑧 ∈ V
45 fveq2 6344 . . . . . . . . 9 (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
46 id 22 . . . . . . . . 9 (𝑥 = 𝑧𝑥 = 𝑧)
4745, 46eqeq12d 2767 . . . . . . . 8 (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
48 sseq2 3760 . . . . . . . 8 (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
49 eqeq1 2756 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
5049notbid 307 . . . . . . . . 9 (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
5150ralbidv 3116 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5247, 48, 513anbi123d 1540 . . . . . . 7 (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
5344, 52elab 3482 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
54 df-3an 1074 . . . . . 6 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)) ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5553, 54bitri 264 . . . . 5 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5655notbii 309 . . . 4 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5743, 56syl6ibr 242 . . 3 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
5857ralrimiv 3095 . 2 (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
59 cardon 8952 . . . . . 6 (card‘𝑥) ∈ On
60 eleq1 2819 . . . . . 6 ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
6159, 60mpbii 223 . . . . 5 ((card‘𝑥) = 𝑥𝑥 ∈ On)
62613ad2ant1 1127 . . . 4 (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
6362abssi 3810 . . 3 {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
64 oneqmini 5929 . . 3 ({𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On → (((ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}) → (ℵ‘𝐴) = {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
6563, 64ax-mp 5 . 2 (((ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ∧ ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}) → (ℵ‘𝐴) = {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
6622, 58, 65syl2anc 696 1 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  {cab 2738  wral 3042  wrex 3043  wss 3707   cint 4619  Oncon0 5876  cfv 6041  ωcom 7222  cardccrd 8943  cale 8944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-reg 8654  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8572  df-har 8620  df-card 8947  df-aleph 8948
This theorem is referenced by: (None)
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