![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alephord2 | Structured version Visualization version GIF version |
Description: Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.) |
Ref | Expression |
---|---|
alephord2 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (ℵ‘𝐴) ∈ (ℵ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephord 9211 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵))) | |
2 | alephon 9205 | . . . 4 ⊢ (ℵ‘𝐴) ∈ On | |
3 | alephon 9205 | . . . . 5 ⊢ (ℵ‘𝐵) ∈ On | |
4 | onenon 9088 | . . . . 5 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (ℵ‘𝐵) ∈ dom card |
6 | cardsdomel 9113 | . . . 4 ⊢ (((ℵ‘𝐴) ∈ On ∧ (ℵ‘𝐵) ∈ dom card) → ((ℵ‘𝐴) ≺ (ℵ‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(ℵ‘𝐵)))) | |
7 | 2, 5, 6 | mp2an 685 | . . 3 ⊢ ((ℵ‘𝐴) ≺ (ℵ‘𝐵) ↔ (ℵ‘𝐴) ∈ (card‘(ℵ‘𝐵))) |
8 | alephcard 9206 | . . . 4 ⊢ (card‘(ℵ‘𝐵)) = (ℵ‘𝐵) | |
9 | 8 | eleq2i 2898 | . . 3 ⊢ ((ℵ‘𝐴) ∈ (card‘(ℵ‘𝐵)) ↔ (ℵ‘𝐴) ∈ (ℵ‘𝐵)) |
10 | 7, 9 | bitri 267 | . 2 ⊢ ((ℵ‘𝐴) ≺ (ℵ‘𝐵) ↔ (ℵ‘𝐴) ∈ (ℵ‘𝐵)) |
11 | 1, 10 | syl6bb 279 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (ℵ‘𝐴) ∈ (ℵ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 class class class wbr 4873 dom cdm 5342 Oncon0 5963 ‘cfv 6123 ≺ csdm 8221 cardccrd 9074 ℵcale 9075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-oi 8684 df-har 8732 df-card 9078 df-aleph 9079 |
This theorem is referenced by: alephord2i 9213 alephord3 9214 alephiso 9234 alephval3 9246 |
Copyright terms: Public domain | W3C validator |