![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alephsson | Structured version Visualization version GIF version |
Description: The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
alephsson | ⊢ ran ℵ ⊆ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinfcard 10163 | . . 3 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ) | |
2 | cardon 10015 | . . . . 5 ⊢ (card‘𝑥) ∈ On | |
3 | eleq1 2832 | . . . . 5 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On)) | |
4 | 2, 3 | mpbii 233 | . . . 4 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On) |
5 | 4 | adantl 481 | . . 3 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) → 𝑥 ∈ On) |
6 | 1, 5 | sylbir 235 | . 2 ⊢ (𝑥 ∈ ran ℵ → 𝑥 ∈ On) |
7 | 6 | ssriv 4012 | 1 ⊢ ran ℵ ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ran crn 5701 Oncon0 6397 ‘cfv 6575 ωcom 7905 cardccrd 10006 ℵcale 10007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-oi 9581 df-har 9628 df-card 10010 df-aleph 10011 |
This theorem is referenced by: unialeph 10172 alephsmo 10173 alephfplem3 10177 alephfp 10179 alephfp2 10180 |
Copyright terms: Public domain | W3C validator |