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Mirrors > Home > MPE Home > Th. List > aleph1 | Structured version Visualization version GIF version |
Description: The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.) |
Ref | Expression |
---|---|
aleph1 | ⊢ (ℵ‘1o) ≼ (2o ↑m (ℵ‘∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8464 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | fveq2i 6887 | . 2 ⊢ (ℵ‘1o) = (ℵ‘suc ∅) |
3 | alephsucpw 10564 | . . 3 ⊢ (ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) | |
4 | fvex 6897 | . . . . 5 ⊢ (ℵ‘∅) ∈ V | |
5 | 4 | pw2en 9078 | . . . 4 ⊢ 𝒫 (ℵ‘∅) ≈ (2o ↑m (ℵ‘∅)) |
6 | domen2 9119 | . . . 4 ⊢ (𝒫 (ℵ‘∅) ≈ (2o ↑m (ℵ‘∅)) → ((ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) ↔ (ℵ‘suc ∅) ≼ (2o ↑m (ℵ‘∅)))) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) ↔ (ℵ‘suc ∅) ≼ (2o ↑m (ℵ‘∅))) |
8 | 3, 7 | mpbi 229 | . 2 ⊢ (ℵ‘suc ∅) ≼ (2o ↑m (ℵ‘∅)) |
9 | 2, 8 | eqbrtri 5162 | 1 ⊢ (ℵ‘1o) ≼ (2o ↑m (ℵ‘∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∅c0 4317 𝒫 cpw 4597 class class class wbr 5141 suc csuc 6359 ‘cfv 6536 (class class class)co 7404 1oc1o 8457 2oc2o 8458 ↑m cmap 8819 ≈ cen 8935 ≼ cdom 8936 ℵcale 9930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-ac2 10457 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-har 9551 df-card 9933 df-aleph 9934 df-ac 10110 |
This theorem is referenced by: (None) |
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