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Theorem alephsuc3 10577
Description: An alternate representation of a successor aleph. Compare alephsuc 10065 and alephsuc2 10077. Equality can be obtained by taking the card of the right-hand side then using alephcard 10067 and carden 10548. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
alephsuc3 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) β‰ˆ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ (β„΅β€˜π΄)})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem alephsuc3
StepHypRef Expression
1 alephsuc2 10077 . . . . 5 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = {π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)})
2 alephcard 10067 . . . . . . 7 (cardβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄)
3 alephon 10066 . . . . . . . . 9 (β„΅β€˜π΄) ∈ On
4 onenon 9946 . . . . . . . . 9 ((β„΅β€˜π΄) ∈ On β†’ (β„΅β€˜π΄) ∈ dom card)
53, 4ax-mp 5 . . . . . . . 8 (β„΅β€˜π΄) ∈ dom card
6 cardval2 9988 . . . . . . . 8 ((β„΅β€˜π΄) ∈ dom card β†’ (cardβ€˜(β„΅β€˜π΄)) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)})
75, 6ax-mp 5 . . . . . . 7 (cardβ€˜(β„΅β€˜π΄)) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)}
82, 7eqtr3i 2756 . . . . . 6 (β„΅β€˜π΄) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)}
98a1i 11 . . . . 5 (𝐴 ∈ On β†’ (β„΅β€˜π΄) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)})
101, 9difeq12d 4118 . . . 4 (𝐴 ∈ On β†’ ((β„΅β€˜suc 𝐴) βˆ– (β„΅β€˜π΄)) = ({π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)} βˆ– {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)}))
11 difrab 4303 . . . . 5 ({π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)} βˆ– {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)}) = {π‘₯ ∈ On ∣ (π‘₯ β‰Ό (β„΅β€˜π΄) ∧ Β¬ π‘₯ β‰Ί (β„΅β€˜π΄))}
12 bren2 8981 . . . . . 6 (π‘₯ β‰ˆ (β„΅β€˜π΄) ↔ (π‘₯ β‰Ό (β„΅β€˜π΄) ∧ Β¬ π‘₯ β‰Ί (β„΅β€˜π΄)))
1312rabbii 3432 . . . . 5 {π‘₯ ∈ On ∣ π‘₯ β‰ˆ (β„΅β€˜π΄)} = {π‘₯ ∈ On ∣ (π‘₯ β‰Ό (β„΅β€˜π΄) ∧ Β¬ π‘₯ β‰Ί (β„΅β€˜π΄))}
1411, 13eqtr4i 2757 . . . 4 ({π‘₯ ∈ On ∣ π‘₯ β‰Ό (β„΅β€˜π΄)} βˆ– {π‘₯ ∈ On ∣ π‘₯ β‰Ί (β„΅β€˜π΄)}) = {π‘₯ ∈ On ∣ π‘₯ β‰ˆ (β„΅β€˜π΄)}
1510, 14eqtr2di 2783 . . 3 (𝐴 ∈ On β†’ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ (β„΅β€˜π΄)} = ((β„΅β€˜suc 𝐴) βˆ– (β„΅β€˜π΄)))
16 alephon 10066 . . . . 5 (β„΅β€˜suc 𝐴) ∈ On
17 onenon 9946 . . . . 5 ((β„΅β€˜suc 𝐴) ∈ On β†’ (β„΅β€˜suc 𝐴) ∈ dom card)
1816, 17mp1i 13 . . . 4 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) ∈ dom card)
19 onsucb 7802 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20 alephgeom 10079 . . . . . 6 (suc 𝐴 ∈ On ↔ Ο‰ βŠ† (β„΅β€˜suc 𝐴))
2119, 20bitri 275 . . . . 5 (𝐴 ∈ On ↔ Ο‰ βŠ† (β„΅β€˜suc 𝐴))
22 fvex 6898 . . . . . 6 (β„΅β€˜suc 𝐴) ∈ V
23 ssdomg 8998 . . . . . 6 ((β„΅β€˜suc 𝐴) ∈ V β†’ (Ο‰ βŠ† (β„΅β€˜suc 𝐴) β†’ Ο‰ β‰Ό (β„΅β€˜suc 𝐴)))
2422, 23ax-mp 5 . . . . 5 (Ο‰ βŠ† (β„΅β€˜suc 𝐴) β†’ Ο‰ β‰Ό (β„΅β€˜suc 𝐴))
2521, 24sylbi 216 . . . 4 (𝐴 ∈ On β†’ Ο‰ β‰Ό (β„΅β€˜suc 𝐴))
26 alephordilem1 10070 . . . 4 (𝐴 ∈ On β†’ (β„΅β€˜π΄) β‰Ί (β„΅β€˜suc 𝐴))
27 infdif 10206 . . . 4 (((β„΅β€˜suc 𝐴) ∈ dom card ∧ Ο‰ β‰Ό (β„΅β€˜suc 𝐴) ∧ (β„΅β€˜π΄) β‰Ί (β„΅β€˜suc 𝐴)) β†’ ((β„΅β€˜suc 𝐴) βˆ– (β„΅β€˜π΄)) β‰ˆ (β„΅β€˜suc 𝐴))
2818, 25, 26, 27syl3anc 1368 . . 3 (𝐴 ∈ On β†’ ((β„΅β€˜suc 𝐴) βˆ– (β„΅β€˜π΄)) β‰ˆ (β„΅β€˜suc 𝐴))
2915, 28eqbrtrd 5163 . 2 (𝐴 ∈ On β†’ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ (β„΅β€˜π΄)} β‰ˆ (β„΅β€˜suc 𝐴))
3029ensymd 9003 1 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) β‰ˆ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ (β„΅β€˜π΄)})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   βˆ– cdif 3940   βŠ† wss 3943   class class class wbr 5141  dom cdm 5669  Oncon0 6358  suc csuc 6360  β€˜cfv 6537  Ο‰com 7852   β‰ˆ cen 8938   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-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-dju 9898  df-card 9936  df-aleph 9937
This theorem is referenced by: (None)
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