Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9376 | . . . . 5 ⊢ (card‘(card‘𝐴)) = (card‘𝐴) | |
2 | cardom 9403 | . . . . . . 7 ⊢ (card‘ω) = ω | |
3 | simpr 485 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ≼ 𝐴) | |
4 | omelon 9097 | . . . . . . . . . 10 ⊢ ω ∈ On | |
5 | onenon 9366 | . . . . . . . . . 10 ⊢ (ω ∈ On → ω ∈ dom card) | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . 9 ⊢ ω ∈ dom card |
7 | simpl 483 | . . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card) | |
8 | carddom2 9394 | . . . . . . . . 9 ⊢ ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴)) | |
9 | 6, 7, 8 | sylancr 587 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((card‘ω) ⊆ (card‘𝐴) ↔ ω ≼ 𝐴)) |
10 | 3, 9 | mpbird 258 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘ω) ⊆ (card‘𝐴)) |
11 | 2, 10 | eqsstrrid 4013 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ω ⊆ (card‘𝐴)) |
12 | cardalephex 9504 | . . . . . 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 234 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥)) |
15 | eqcom 2825 | . . . . 5 ⊢ ((card‘𝐴) = (ℵ‘𝑥) ↔ (ℵ‘𝑥) = (card‘𝐴)) | |
16 | 15 | rexbii 3244 | . . . 4 ⊢ (∃𝑥 ∈ On (card‘𝐴) = (ℵ‘𝑥) ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)) |
17 | 14, 16 | sylib 219 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘𝐴)) |
18 | alephfnon 9479 | . . . 4 ⊢ ℵ Fn On | |
19 | fvelrnb 6719 | . . . 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 235 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ∈ ran ℵ) |
22 | cardid2 9370 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
23 | 22 | adantr 481 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (card‘𝐴) ≈ 𝐴) |
24 | breq1 5060 | . . 3 ⊢ (𝑥 = (card‘𝐴) → (𝑥 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴)) | |
25 | 24 | rspcev 3620 | . 2 ⊢ (((card‘𝐴) ∈ ran ℵ ∧ (card‘𝐴) ≈ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴) |
26 | 21, 23, 25 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 dom cdm 5548 ran crn 5549 Oncon0 6184 Fn wfn 6343 ‘cfv 6348 ωcom 7569 ≈ cen 8494 ≼ cdom 8495 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-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: (None) |
Copyright terms: Public domain | W3C validator |