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Theorem alephf1 10125
Description: The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 10143. (Contributed by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephf1 ℵ:On–1-1→On

Proof of Theorem alephf1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfnon 10105 . . 3 ℵ Fn On
2 alephon 10109 . . . 4 (ℵ‘𝑥) ∈ On
32rgenw 3065 . . 3 𝑥 ∈ On (ℵ‘𝑥) ∈ On
4 ffnfv 7139 . . 3 (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On))
51, 3, 4mpbir2an 711 . 2 ℵ:On⟶On
6 aleph11 10124 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) ↔ 𝑥 = 𝑦))
76biimpd 229 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦))
87rgen2 3199 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)
9 dff13 7275 . 2 (ℵ:On–1-1→On ↔ (ℵ:On⟶On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)))
105, 8, 9mpbir2an 711 1 ℵ:On–1-1→On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Oncon0 6384   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  cale 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980
This theorem is referenced by: (None)
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