Theorem alephf1 9510

Description: The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 9528. (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.

Step	Hyp	Ref	Expression
1		alephfnon 9490	. . 3 ⊢ ℵ Fn On
2		alephon 9494	. . . 4 ⊢ (ℵ‘𝑥) ∈ On
3	2	rgenw 3150	. . 3 ⊢ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On
4		ffnfv 6881	. . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On))
5	1, 3, 4	mpbir2an 709	. 2 ⊢ ℵ:On⟶On
6		aleph11 9509	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) ↔ 𝑥 = 𝑦))
7	6	biimpd 231	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦))
8	7	rgen2 3203	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)
9		dff13 7012	. 2 ⊢ (ℵ:On–1-1→On ↔ (ℵ:On⟶On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)))
10	5, 8, 9	mpbir2an 709	1 ⊢ ℵ:On–1-1→On

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Oncon0 6190   Fn wfn 6349  ⟶wf 6350  –1-1→wf1 6351  ‘cfv 6354  ℵcale 9364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-oi 8973  df-har 9021  df-card 9367  df-aleph 9368

This theorem is referenced by: (None)