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Theorem alephval3 10148
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 10108 . . . 4 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3 alephgeom 10120 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
43biimpi 216 . . 3 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5 alephord2i 10115 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6 elirr 9635 . . . . . . 7 ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7 eleq2 2828 . . . . . . 7 ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
86, 7mtbiri 327 . . . . . 6 ((ℵ‘𝐴) = (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (ℵ‘𝐴))
98con2i 139 . . . . 5 ((ℵ‘𝑦) ∈ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
105, 9syl6 35 . . . 4 (𝐴 ∈ On → (𝑦𝐴 → ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1110ralrimiv 3143 . . 3 (𝐴 ∈ On → ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
12 fvex 6920 . . . 4 (ℵ‘𝐴) ∈ V
13 fveq2 6907 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14 id 22 . . . . . 6 (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
1513, 14eqeq12d 2751 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16 sseq2 4022 . . . . 5 (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17 eqeq1 2739 . . . . . . 7 (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
1817notbid 318 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1918ralbidv 3176 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
2015, 16, 193anbi123d 1435 . . . 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 3681 . . 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 1342 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23 eleq1 2827 . . . . . . . . . . . . . . 15 (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
24 alephord2 10114 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
2524bicomd 223 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2623, 25sylan9bbr 510 . . . . . . . . . . . . . 14 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2726biimpcd 249 . . . . . . . . . . . . 13 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦𝐴))
28 simpr 484 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
2927, 28jca2 513 . . . . . . . . . . . 12 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3029exp4c 432 . . . . . . . . . . 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 421 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3332reximdv2 3162 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦)))
34 cardalephex 10128 . . . . . . . . 9 (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
3534biimpac 478 . . . . . . . 8 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
3633, 35impel 505 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦))
37 dfrex2 3071 . . . . . . 7 (∃𝑦𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
3836, 37sylib 218 . . . . . 6 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39 nan 830 . . . . . 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 231 . . . . 5 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
4140ex 412 . . . 4 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
42 vex 3482 . . . . . . 7 𝑧 ∈ V
43 fveq2 6907 . . . . . . . . 9 (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
44 id 22 . . . . . . . . 9 (𝑥 = 𝑧𝑥 = 𝑧)
4543, 44eqeq12d 2751 . . . . . . . 8 (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
46 sseq2 4022 . . . . . . . 8 (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
47 eqeq1 2739 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
4847notbid 318 . . . . . . . . 9 (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
4948ralbidv 3176 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5045, 46, 493anbi123d 1435 . . . . . . 7 (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
5142, 50elab 3681 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
52 df-3an 1088 . . . . . 6 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)) ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5351, 52bitri 275 . . . . 5 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5453notbii 320 . . . 4 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5541, 54imbitrrdi 252 . . 3 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
5655ralrimiv 3143 . 2 (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
57 cardon 9982 . . . . . 6 (card‘𝑥) ∈ On
58 eleq1 2827 . . . . . 6 ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
5957, 58mpbii 233 . . . . 5 ((card‘𝑥) = 𝑥𝑥 ∈ On)
60593ad2ant1 1132 . . . 4 (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
6160abssi 4080 . . 3 {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
62 oneqmini 6438 . . 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 584 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 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963   cint 4951  Oncon0 6386  cfv 6563  ωcom 7887  cardccrd 9973  cale 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-card 9977  df-aleph 9978
This theorem is referenced by: (None)
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