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Mirrors > Home > MPE Home > Th. List > alephislim | Structured version Visualization version GIF version |
Description: Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
alephislim | ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephgeom 9496 | . 2 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
2 | cardlim 9389 | . . 3 ⊢ (ω ⊆ (card‘(ℵ‘𝐴)) ↔ Lim (card‘(ℵ‘𝐴))) | |
3 | alephcard 9484 | . . . 4 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴) | |
4 | 3 | sseq2i 3993 | . . 3 ⊢ (ω ⊆ (card‘(ℵ‘𝐴)) ↔ ω ⊆ (ℵ‘𝐴)) |
5 | limeq 6196 | . . . 4 ⊢ ((card‘(ℵ‘𝐴)) = (ℵ‘𝐴) → (Lim (card‘(ℵ‘𝐴)) ↔ Lim (ℵ‘𝐴))) | |
6 | 3, 5 | ax-mp 5 | . . 3 ⊢ (Lim (card‘(ℵ‘𝐴)) ↔ Lim (ℵ‘𝐴)) |
7 | 2, 4, 6 | 3bitr3i 302 | . 2 ⊢ (ω ⊆ (ℵ‘𝐴) ↔ Lim (ℵ‘𝐴)) |
8 | 1, 7 | bitri 276 | 1 ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 Oncon0 6184 Lim wlim 6185 ‘cfv 6348 ωcom 7569 cardccrd 9352 ℵcale 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-har 9010 df-card 9356 df-aleph 9357 |
This theorem is referenced by: alephreg 9992 pwcfsdom 9993 |
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