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Theorem alephf1 9021
Description: The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 9039. (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 9001 . . 3 ℵ Fn On
2 alephon 9005 . . . 4 (ℵ‘𝑥) ∈ On
32rgenw 3026 . . 3 𝑥 ∈ On (ℵ‘𝑥) ∈ On
4 ffnfv 6503 . . 3 (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On))
51, 3, 4mpbir2an 993 . 2 ℵ:On⟶On
6 aleph11 9020 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) ↔ 𝑥 = 𝑦))
76biimpd 219 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦))
87rgen2a 3079 . 2 𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)
9 dff13 6627 . 2 (ℵ:On–1-1→On ↔ (ℵ:On⟶On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)))
105, 8, 9mpbir2an 993 1 ℵ:On–1-1→On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wral 3014  Oncon0 5836   Fn wfn 5996  wf 5997  1-1wf1 5998  cfv 6001  cale 8875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-oi 8531  df-har 8579  df-card 8878  df-aleph 8879
This theorem is referenced by: (None)
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