Theorem alephf1 10068

Description: The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 10086. (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 10048	. . 3 ⊢ ℵ Fn On
2		alephon 10052	. . . 4 ⊢ (ℵ‘𝑥) ∈ On
3	2	rgenw 3081	. . 3 ⊢ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On
4		ffnfv 7114	. . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On))
5	1, 3, 4	mpbir2an 723	. 2 ⊢ ℵ:On⟶On
6		aleph11 10067	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) ↔ 𝑥 = 𝑦))
7	6	biimpd 232	. . 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 7252	. 2 ⊢ (ℵ:On–1-1→On ↔ (ℵ:On⟶On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)))
10	5, 8, 9	mpbir2an 723	1 ⊢ ℵ:On–1-1→On

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  Oncon0 6360   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  ℵcale 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-oi 9471  df-har 9518  df-card 9924  df-aleph 9925

This theorem is referenced by: (None)