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Theorem alephval3 9014
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 8974 . . . 4 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3 alephgeom 8986 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
43biimpi 206 . . 3 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5 alephord2i 8981 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6 elirr 8586 . . . . . . 7 ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7 eleq2 2760 . . . . . . 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 3035 . . 3 (𝐴 ∈ On → ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
12 fvex 6282 . . . 4 (ℵ‘𝐴) ∈ V
13 fveq2 6272 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14 id 22 . . . . . 6 (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
1513, 14eqeq12d 2707 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16 sseq2 3701 . . . . 5 (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17 eqeq1 2696 . . . . . . 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 3056 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
2015, 16, 193anbi123d 1480 . . . 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 3423 . . 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 1319 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23 cardalephex 8994 . . . . . . . . . 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 2759 . . . . . . . . . . . . . . . 16 (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
26 alephord2 8980 . . . . . . . . . . . . . . . . 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 3084 . . . . . . . . 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 3066 . . . . . . 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 3275 . . . . . . 7 𝑧 ∈ V
45 fveq2 6272 . . . . . . . . 9 (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
46 id 22 . . . . . . . . 9 (𝑥 = 𝑧𝑥 = 𝑧)
4745, 46eqeq12d 2707 . . . . . . . 8 (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
48 sseq2 3701 . . . . . . . 8 (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
49 eqeq1 2696 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
5049notbid 307 . . . . . . . . 9 (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
5150ralbidv 3056 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5247, 48, 513anbi123d 1480 . . . . . . 7 (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
5344, 52elab 3423 . . . . . 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 3035 . 2 (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
59 cardon 8851 . . . . . 6 (card‘𝑥) ∈ On
60 eleq1 2759 . . . . . 6 ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
6159, 60mpbii 223 . . . . 5 ((card‘𝑥) = 𝑥𝑥 ∈ On)
62613ad2ant1 1125 . . . 4 (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
6362abssi 3751 . . 3 {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
64 oneqmini 5857 . . 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 1564  wcel 2071  {cab 2678  wral 2982  wrex 2983  wss 3648   cint 4551  Oncon0 5804  cfv 5969  ωcom 7150  cardccrd 8842  cale 8843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034  ax-reg 8581  ax-inf2 8619
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rmo 2990  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-uni 4513  df-int 4552  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-se 5146  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-pred 5761  df-ord 5807  df-on 5808  df-lim 5809  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-isom 5978  df-riota 6694  df-om 7151  df-wrecs 7495  df-recs 7556  df-rdg 7594  df-er 7830  df-en 8041  df-dom 8042  df-sdom 8043  df-fin 8044  df-oi 8499  df-har 8547  df-card 8846  df-aleph 8847
This theorem is referenced by: (None)
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