Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > alephordilem1 | Structured version Visualization version GIF version |
Description: Lemma for alephordi 9494. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
alephordilem1 | ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephon 9489 | . . 3 ⊢ (ℵ‘𝐴) ∈ On | |
2 | onenon 9372 | . . 3 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
3 | harsdom 9418 | . . 3 ⊢ ((ℵ‘𝐴) ∈ dom card → (ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴)) |
5 | alephsuc 9488 | . 2 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴))) | |
6 | 4, 5 | breqtrrid 5096 | 1 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5058 dom cdm 5549 Oncon0 6185 suc csuc 6187 ‘cfv 6349 ≺ csdm 8502 harchar 9014 cardccrd 9358 ℵcale 9359 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-oi 8968 df-har 9016 df-card 9362 df-aleph 9363 |
This theorem is referenced by: alephordi 9494 alephsucdom 9499 alephsuc3 9996 alephreg 9998 gchaleph 10087 |
Copyright terms: Public domain | W3C validator |