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Mirrors > Home > MPE Home > Th. List > alephf1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
alephf1 | ⊢ ℵ:On–1-1→On |
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 |
Colors of variables: wff setvar class |
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) |
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