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Theorem alephsuc2 10077
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9541 function by transfinite recursion, starting from Ο‰. Using this theorem we could define the aleph function with {𝑧 ∈ On ∣ 𝑧 β‰Ό π‘₯} in place of ∩ {𝑧 ∈ On ∣ π‘₯ β‰Ί 𝑧} in df-aleph 9937. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephsuc2 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem alephsuc2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 alephon 10066 . . . . . . 7 (β„΅β€˜suc 𝐴) ∈ On
21oneli 6472 . . . . . 6 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ On)
3 alephcard 10067 . . . . . . . . 9 (cardβ€˜(β„΅β€˜suc 𝐴)) = (β„΅β€˜suc 𝐴)
4 iscard 9972 . . . . . . . . 9 ((cardβ€˜(β„΅β€˜suc 𝐴)) = (β„΅β€˜suc 𝐴) ↔ ((β„΅β€˜suc 𝐴) ∈ On ∧ βˆ€π‘¦ ∈ (β„΅β€˜suc 𝐴)𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
53, 4mpbi 229 . . . . . . . 8 ((β„΅β€˜suc 𝐴) ∈ On ∧ βˆ€π‘¦ ∈ (β„΅β€˜suc 𝐴)𝑦 β‰Ί (β„΅β€˜suc 𝐴))
65simpri 485 . . . . . . 7 βˆ€π‘¦ ∈ (β„΅β€˜suc 𝐴)𝑦 β‰Ί (β„΅β€˜suc 𝐴)
76rspec 3241 . . . . . 6 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ 𝑦 β‰Ί (β„΅β€˜suc 𝐴))
82, 7jca 511 . . . . 5 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
9 sdomel 9126 . . . . . . 7 ((𝑦 ∈ On ∧ (β„΅β€˜suc 𝐴) ∈ On) β†’ (𝑦 β‰Ί (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴)))
101, 9mpan2 688 . . . . . 6 (𝑦 ∈ On β†’ (𝑦 β‰Ί (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴)))
1110imp 406 . . . . 5 ((𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴))
128, 11impbii 208 . . . 4 (𝑦 ∈ (β„΅β€˜suc 𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
13 breq1 5144 . . . . 5 (π‘₯ = 𝑦 β†’ (π‘₯ β‰Ί (β„΅β€˜suc 𝐴) ↔ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
1413elrab 3678 . . . 4 (𝑦 ∈ {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
1512, 14bitr4i 278 . . 3 (𝑦 ∈ (β„΅β€˜suc 𝐴) ↔ 𝑦 ∈ {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)})
1615eqriv 2723 . 2 (β„΅β€˜suc 𝐴) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)}
17 alephsucdom 10076 . . 3 (𝐴 ∈ On β†’ (π‘₯ β‰Ό (β„΅β€˜π΄) ↔ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)))
1817rabbidv 3434 . 2 (𝐴 ∈ On β†’ {π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)} = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)})
1916, 18eqtr4id 2785 1 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   class class class wbr 5141  Oncon0 6358  suc csuc 6360  β€˜cfv 6537   β‰Ό cdom 8939   β‰Ί csdm 8940  cardccrd 9932  β„΅cale 9933
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937
This theorem is referenced by:  alephsuc3  10577
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