Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > alephsdom | Structured version Visualization version GIF version |
Description: If an ordinal is smaller than an initial ordinal, it is strictly dominated by it. (Contributed by Jeff Hankins, 24-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
alephsdom | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | |
2 | alephon 9826 | . . . 4 ⊢ (ℵ‘𝐵) ∈ On | |
3 | onenon 9708 | . . . 4 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐵) ∈ dom card |
5 | cardsdomel 9733 | . . 3 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐵) ∈ dom card) → (𝐴 ≺ (ℵ‘𝐵) ↔ 𝐴 ∈ (card‘(ℵ‘𝐵)))) | |
6 | 1, 4, 5 | sylancl 586 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≺ (ℵ‘𝐵) ↔ 𝐴 ∈ (card‘(ℵ‘𝐵)))) |
7 | alephcard 9827 | . . 3 ⊢ (card‘(ℵ‘𝐵)) = (ℵ‘𝐵) | |
8 | 7 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ (card‘(ℵ‘𝐵)) ↔ 𝐴 ∈ (ℵ‘𝐵)) |
9 | 6, 8 | bitr2di 288 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2110 class class class wbr 5079 dom cdm 5590 Oncon0 6265 ‘cfv 6432 ≺ csdm 8715 cardccrd 9694 ℵcale 9695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-oi 9247 df-har 9294 df-card 9698 df-aleph 9699 |
This theorem is referenced by: alephdom2 9844 |
Copyright terms: Public domain | W3C validator |