Theorem alephf1 10007

Description: The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 10025. (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 9987	. . 3 ⊢ ℵ Fn On
2		alephon 9991	. . . 4 ⊢ (ℵ‘𝑥) ∈ On
3	2	rgenw 3056	. . 3 ⊢ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On
4		ffnfv 7072	. . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On))
5	1, 3, 4	mpbir2an 712	. 2 ⊢ ℵ:On⟶On
6		aleph11 10006	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) ↔ 𝑥 = 𝑦))
7	6	biimpd 229	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦))
8	7	rgen2 3178	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)
9		dff13 7209	. 2 ⊢ (ℵ:On–1-1→On ↔ (ℵ:On⟶On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On ((ℵ‘𝑥) = (ℵ‘𝑦) → 𝑥 = 𝑦)))
10	5, 8, 9	mpbir2an 712	1 ⊢ ℵ:On–1-1→On

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Oncon0 6324   Fn wfn 6494  ⟶wf 6495  –1-1→wf1 6496  ‘cfv 6499  ℵcale 9860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-oi 9425  df-har 9472  df-card 9863  df-aleph 9864

This theorem is referenced by: (None)