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Mirrors > Home > MPE Home > Th. List > infenaleph | Structured version Visualization version GIF version |
Description: An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
infenaleph | ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardidm 9382 | . . . . 5 ⊢ (card‘(card‘𝐴)) = (card‘𝐴) | |
2 | cardom 9409 | . . . . . . 7 ⊢ (card‘ω) = ω | |
3 | simpr 487 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴) | |
4 | omelon 9103 | . . . . . . . . . 10 ⊢ ω ∈ On | |
5 | onenon 9372 | . . . . . . . . . 10 ⊢ (ω ∈ On → ω ∈ dom card) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . 9 ⊢ ω ∈ dom card |
7 | simpl 485 | . . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card) | |
8 | carddom2 9400 | . . . . . . . . 9 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴)) | |
9 | 6, 7, 8 | sylancr 589 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴)) |
10 | 3, 9 | mpbird 259 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴)) |
11 | 2, 10 | eqsstrrid 4015 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴)) |
12 | cardalephex 9510 | . . . . . 6 ⊢ (ω ⊆ (card‘𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘(card‘𝐴)) = (card‘𝐴) ↔ ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥))) |
14 | 1, 13 | mpbii 235 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)) |
15 | eqcom 2828 | . . . . 5 ⊢ ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴)) | |
16 | 15 | rexbii 3247 | . . . 4 ⊢ (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)) |
17 | 14, 16 | sylib 220 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)) |
18 | alephfnon 9485 | . . . 4 ⊢ ℵ Fn On | |
19 | fvelrnb 6720 | . . . 4 ⊢ (ℵ Fn On → ((card‘𝐴) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴))) | |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ ((card‘𝐴) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)) |
21 | 17, 20 | sylibr 236 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ) |
22 | cardid2 9376 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
23 | 22 | adantr 483 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴) |
24 | breq1 5061 | . . 3 ⊢ (𝑥 = (card‘𝐴) → (𝑥 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴)) | |
25 | 24 | rspcev 3622 | . 2 ⊢ (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴) |
26 | 21, 23, 25 | syl2anc 586 | 1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 dom cdm 5549 ran crn 5550 Oncon0 6185 Fn wfn 6344 ‘cfv 6349 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 cardccrd 9358 ℵcale 9359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-har 9016 df-card 9362 df-aleph 9363 |
This theorem is referenced by: (None) |
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