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Theorem alephval3 9571
 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 9531 . . . 4 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3 alephgeom 9543 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
43biimpi 219 . . 3 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5 alephord2i 9538 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6 elirr 9095 . . . . . . 7 ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7 eleq2 2841 . . . . . . 7 ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
86, 7mtbiri 331 . . . . . 6 ((ℵ‘𝐴) = (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (ℵ‘𝐴))
98con2i 141 . . . . 5 ((ℵ‘𝑦) ∈ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
105, 9syl6 35 . . . 4 (𝐴 ∈ On → (𝑦𝐴 → ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1110ralrimiv 3113 . . 3 (𝐴 ∈ On → ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
12 fvex 6672 . . . 4 (ℵ‘𝐴) ∈ V
13 fveq2 6659 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14 id 22 . . . . . 6 (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
1513, 14eqeq12d 2775 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16 sseq2 3919 . . . . 5 (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17 eqeq1 2763 . . . . . . 7 (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
1817notbid 322 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1918ralbidv 3127 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
2015, 16, 193anbi123d 1434 . . . 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 3589 . . 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 1341 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23 eleq1 2840 . . . . . . . . . . . . . . 15 (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
24 alephord2 9537 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
2524bicomd 226 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2623, 25sylan9bbr 515 . . . . . . . . . . . . . 14 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2726biimpcd 252 . . . . . . . . . . . . 13 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦𝐴))
28 simpr 489 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
2927, 28jca2 518 . . . . . . . . . . . 12 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3029exp4c 437 . . . . . . . . . . 11 (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝐴 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))))
3130com3r 87 . . . . . . . . . 10 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → (𝑦 ∈ On → (𝑧 = (ℵ‘𝑦) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))))
3231imp4b 426 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3332reximdv2 3196 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦)))
34 cardalephex 9551 . . . . . . . . 9 (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
3534biimpac 483 . . . . . . . 8 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
3633, 35impel 510 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦))
37 dfrex2 3167 . . . . . . 7 (∃𝑦𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
3836, 37sylib 221 . . . . . 6 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39 nan 829 . . . . . 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)))
4038, 39mpbir 234 . . . . 5 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
4140ex 417 . . . 4 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
42 vex 3414 . . . . . . 7 𝑧 ∈ V
43 fveq2 6659 . . . . . . . . 9 (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
44 id 22 . . . . . . . . 9 (𝑥 = 𝑧𝑥 = 𝑧)
4543, 44eqeq12d 2775 . . . . . . . 8 (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
46 sseq2 3919 . . . . . . . 8 (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
47 eqeq1 2763 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
4847notbid 322 . . . . . . . . 9 (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
4948ralbidv 3127 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5045, 46, 493anbi123d 1434 . . . . . . 7 (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
5142, 50elab 3589 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
52 df-3an 1087 . . . . . 6 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)) ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5351, 52bitri 278 . . . . 5 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5453notbii 324 . . . 4 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5541, 54syl6ibr 255 . . 3 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
5655ralrimiv 3113 . 2 (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
57 cardon 9407 . . . . . 6 (card‘𝑥) ∈ On
58 eleq1 2840 . . . . . 6 ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
5957, 58mpbii 236 . . . . 5 ((card‘𝑥) = 𝑥𝑥 ∈ On)
60593ad2ant1 1131 . . . 4 (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
6160abssi 3975 . . 3 {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
62 oneqmini 6221 . . 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))}))
6361, 62ax-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))})
6422, 56, 63syl2anc 588 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 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  {cab 2736  ∀wral 3071  ∃wrex 3072   ⊆ wss 3859  ∩ cint 4839  Oncon0 6170  ‘cfv 6336  ωcom 7580  cardccrd 9398  ℵcale 9399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-reg 9090  ax-inf2 9138 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-oi 9008  df-har 9055  df-card 9402  df-aleph 9403 This theorem is referenced by: (None)
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