Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iscard3 | Structured version Visualization version GIF version |
Description: Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.) |
Ref | Expression |
---|---|
iscard3 | ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 8770 | . . . . . . . . 9 ⊢ (card‘𝐴) ∈ On | |
2 | eleq1 2689 | . . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 223 | . . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | eloni 5733 | . . . . . . . 8 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → Ord 𝐴) |
6 | ordom 7074 | . . . . . . 7 ⊢ Ord ω | |
7 | ordtri2or 5822 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) | |
8 | 5, 6, 7 | sylancl 694 | . . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) |
9 | 8 | ord 392 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴)) |
10 | isinfcard 8915 | . . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ) | |
11 | 10 | biimpi 206 | . . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ) |
12 | 11 | expcom 451 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 → 𝐴 ∈ ran ℵ)) |
13 | 9, 12 | syld 47 | . . . 4 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ)) |
14 | 13 | orrd 393 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
15 | cardnn 8789 | . . . 4 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
16 | 10 | bicomi 214 | . . . . 5 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴)) |
17 | 16 | simprbi 480 | . . . 4 ⊢ (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴) |
18 | 15, 17 | jaoi 394 | . . 3 ⊢ ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴) |
19 | 14, 18 | impbii 199 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
20 | elun 3753 | . 2 ⊢ (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) | |
21 | 19, 20 | bitr4i 267 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 ran crn 5115 Ord word 5722 Oncon0 5723 ‘cfv 5888 ωcom 7065 cardccrd 8761 ℵcale 8762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-har 8463 df-card 8765 df-aleph 8766 |
This theorem is referenced by: cardnum 8917 carduniima 8919 cardinfima 8920 cfpwsdom 9406 gch2 9497 |
Copyright terms: Public domain | W3C validator |