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Theorem alephsuc2 10103
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9567 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 9963. (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 10092 . . . . . . 7 (β„΅β€˜suc 𝐴) ∈ On
21oneli 6478 . . . . . 6 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ On)
3 alephcard 10093 . . . . . . . . 9 (cardβ€˜(β„΅β€˜suc 𝐴)) = (β„΅β€˜suc 𝐴)
4 iscard 9998 . . . . . . . . 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 484 . . . . . . 7 βˆ€π‘¦ ∈ (β„΅β€˜suc 𝐴)𝑦 β‰Ί (β„΅β€˜suc 𝐴)
76rspec 3238 . . . . . 6 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ 𝑦 β‰Ί (β„΅β€˜suc 𝐴))
82, 7jca 510 . . . . 5 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
9 sdomel 9147 . . . . . . 7 ((𝑦 ∈ On ∧ (β„΅β€˜suc 𝐴) ∈ On) β†’ (𝑦 β‰Ί (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴)))
101, 9mpan2 689 . . . . . 6 (𝑦 ∈ On β†’ (𝑦 β‰Ί (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴)))
1110imp 405 . . . . 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 5146 . . . . 5 (π‘₯ = 𝑦 β†’ (π‘₯ β‰Ί (β„΅β€˜suc 𝐴) ↔ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
1413elrab 3674 . . . 4 (𝑦 ∈ {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
1512, 14bitr4i 277 . . 3 (𝑦 ∈ (β„΅β€˜suc 𝐴) ↔ 𝑦 ∈ {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)})
1615eqriv 2722 . 2 (β„΅β€˜suc 𝐴) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)}
17 alephsucdom 10102 . . 3 (𝐴 ∈ On β†’ (π‘₯ β‰Ό (β„΅β€˜π΄) ↔ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)))
1817rabbidv 3427 . 2 (𝐴 ∈ On β†’ {π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)} = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)})
1916, 18eqtr4id 2784 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 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  {crab 3419   class class class wbr 5143  Oncon0 6364  suc csuc 6366  β€˜cfv 6543   β‰Ό cdom 8960   β‰Ί csdm 8961  cardccrd 9958  β„΅cale 9959
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-har 9580  df-card 9962  df-aleph 9963
This theorem is referenced by:  alephsuc3  10603
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