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Theorem alephsuc2 10023
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9487 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 9883. (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 10012 . . . . . . 7 (β„΅β€˜suc 𝐴) ∈ On
21oneli 6436 . . . . . 6 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ On)
3 alephcard 10013 . . . . . . . . 9 (cardβ€˜(β„΅β€˜suc 𝐴)) = (β„΅β€˜suc 𝐴)
4 iscard 9918 . . . . . . . . 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 487 . . . . . . 7 βˆ€π‘¦ ∈ (β„΅β€˜suc 𝐴)𝑦 β‰Ί (β„΅β€˜suc 𝐴)
76rspec 3236 . . . . . 6 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ 𝑦 β‰Ί (β„΅β€˜suc 𝐴))
82, 7jca 513 . . . . 5 (𝑦 ∈ (β„΅β€˜suc 𝐴) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
9 sdomel 9075 . . . . . . 7 ((𝑦 ∈ On ∧ (β„΅β€˜suc 𝐴) ∈ On) β†’ (𝑦 β‰Ί (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴)))
101, 9mpan2 690 . . . . . 6 (𝑦 ∈ On β†’ (𝑦 β‰Ί (β„΅β€˜suc 𝐴) β†’ 𝑦 ∈ (β„΅β€˜suc 𝐴)))
1110imp 408 . . . . 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 5113 . . . . 5 (π‘₯ = 𝑦 β†’ (π‘₯ β‰Ί (β„΅β€˜suc 𝐴) ↔ 𝑦 β‰Ί (β„΅β€˜suc 𝐴)))
1413elrab 3650 . . . 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 2734 . 2 (β„΅β€˜suc 𝐴) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)}
17 alephsucdom 10022 . . 3 (𝐴 ∈ On β†’ (π‘₯ β‰Ό (β„΅β€˜π΄) ↔ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)))
1817rabbidv 3418 . 2 (𝐴 ∈ On β†’ {π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)} = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜suc 𝐴)})
1916, 18eqtr4id 2796 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410   class class class wbr 5110  Oncon0 6322  suc csuc 6324  β€˜cfv 6501   β‰Ό cdom 8888   β‰Ί csdm 8889  cardccrd 9878  β„΅cale 9879
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-har 9500  df-card 9882  df-aleph 9883
This theorem is referenced by:  alephsuc3  10523
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