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Theorem alephval3 9525
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 9485 . . . 4 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
21a1i 11 . . 3 (𝐴 ∈ On → (card‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3 alephgeom 9497 . . . 4 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
43biimpi 217 . . 3 (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
5 alephord2i 9492 . . . . 5 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
6 elirr 9050 . . . . . . 7 ¬ (ℵ‘𝑦) ∈ (ℵ‘𝑦)
7 eleq2 2906 . . . . . . 7 ((ℵ‘𝐴) = (ℵ‘𝑦) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑦)))
86, 7mtbiri 328 . . . . . 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 3186 . . 3 (𝐴 ∈ On → ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦))
12 fvex 6680 . . . 4 (ℵ‘𝐴) ∈ V
13 fveq2 6667 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (card‘𝑥) = (card‘(ℵ‘𝐴)))
14 id 22 . . . . . 6 (𝑥 = (ℵ‘𝐴) → 𝑥 = (ℵ‘𝐴))
1513, 14eqeq12d 2842 . . . . 5 (𝑥 = (ℵ‘𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)))
16 sseq2 3997 . . . . 5 (𝑥 = (ℵ‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (ℵ‘𝐴)))
17 eqeq1 2830 . . . . . . 7 (𝑥 = (ℵ‘𝐴) → (𝑥 = (ℵ‘𝑦) ↔ (ℵ‘𝐴) = (ℵ‘𝑦)))
1817notbid 319 . . . . . 6 (𝑥 = (ℵ‘𝐴) → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
1918ralbidv 3202 . . . . 5 (𝑥 = (ℵ‘𝐴) → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ (ℵ‘𝐴) = (ℵ‘𝑦)))
2015, 16, 193anbi123d 1429 . . . 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 3671 . . 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 1337 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
23 eleq1 2905 . . . . . . . . . . . . . . 15 (𝑧 = (ℵ‘𝑦) → (𝑧 ∈ (ℵ‘𝐴) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
24 alephord2 9491 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
2524bicomd 224 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((ℵ‘𝑦) ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2623, 25sylan9bbr 511 . . . . . . . . . . . . . 14 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑧 ∈ (ℵ‘𝐴) ↔ 𝑦𝐴))
2726biimpcd 250 . . . . . . . . . . . . 13 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑦𝐴))
28 simpr 485 . . . . . . . . . . . . 13 (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → 𝑧 = (ℵ‘𝑦))
2927, 28jca2 514 . . . . . . . . . . . 12 (𝑧 ∈ (ℵ‘𝐴) → (((𝑦 ∈ On ∧ 𝐴 ∈ On) ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3029exp4c 433 . . . . . . . . . . 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 422 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ((𝑦 ∈ On ∧ 𝑧 = (ℵ‘𝑦)) → (𝑦𝐴𝑧 = (ℵ‘𝑦))))
3332reximdv2 3276 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → (∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦)))
34 cardalephex 9505 . . . . . . . . 9 (ω ⊆ 𝑧 → ((card‘𝑧) = 𝑧 ↔ ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦)))
3534biimpac 479 . . . . . . . 8 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) → ∃𝑦 ∈ On 𝑧 = (ℵ‘𝑦))
3633, 35impel 506 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ∃𝑦𝐴 𝑧 = (ℵ‘𝑦))
37 dfrex2 3244 . . . . . . 7 (∃𝑦𝐴 𝑧 = (ℵ‘𝑦) ↔ ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
3836, 37sylib 219 . . . . . 6 (((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) ∧ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧)) → ¬ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))
39 nan 827 . . . . . 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 232 . . . . 5 ((𝐴 ∈ On ∧ 𝑧 ∈ (ℵ‘𝐴)) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
4140ex 413 . . . 4 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
42 vex 3503 . . . . . . 7 𝑧 ∈ V
43 fveq2 6667 . . . . . . . . 9 (𝑥 = 𝑧 → (card‘𝑥) = (card‘𝑧))
44 id 22 . . . . . . . . 9 (𝑥 = 𝑧𝑥 = 𝑧)
4543, 44eqeq12d 2842 . . . . . . . 8 (𝑥 = 𝑧 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑧) = 𝑧))
46 sseq2 3997 . . . . . . . 8 (𝑥 = 𝑧 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑧))
47 eqeq1 2830 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (ℵ‘𝑦) ↔ 𝑧 = (ℵ‘𝑦)))
4847notbid 319 . . . . . . . . 9 (𝑥 = 𝑧 → (¬ 𝑥 = (ℵ‘𝑦) ↔ ¬ 𝑧 = (ℵ‘𝑦)))
4948ralbidv 3202 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦) ↔ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5045, 46, 493anbi123d 1429 . . . . . . 7 (𝑥 = 𝑧 → (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦))))
5142, 50elab 3671 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
52 df-3an 1083 . . . . . 6 (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧 ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)) ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5351, 52bitri 276 . . . . 5 (𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5453notbii 321 . . . 4 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ↔ ¬ (((card‘𝑧) = 𝑧 ∧ ω ⊆ 𝑧) ∧ ∀𝑦𝐴 ¬ 𝑧 = (ℵ‘𝑦)))
5541, 54syl6ibr 253 . . 3 (𝐴 ∈ On → (𝑧 ∈ (ℵ‘𝐴) → ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))}))
5655ralrimiv 3186 . 2 (𝐴 ∈ On → ∀𝑧 ∈ (ℵ‘𝐴) ¬ 𝑧 ∈ {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
57 cardon 9362 . . . . . 6 (card‘𝑥) ∈ On
58 eleq1 2905 . . . . . 6 ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On))
5957, 58mpbii 234 . . . . 5 ((card‘𝑥) = 𝑥𝑥 ∈ On)
60593ad2ant1 1127 . . . 4 (((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦)) → 𝑥 ∈ On)
6160abssi 4050 . . 3 {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))} ⊆ On
62 oneqmini 6240 . . 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 396  w3a 1081   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  wss 3940   cint 4874  Oncon0 6189  cfv 6352  ωcom 7568  cardccrd 9353  cale 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-reg 9045  ax-inf2 9093
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  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
This theorem is referenced by: (None)
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