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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 9449 | . . . . . . . . 9 ⊢ (card‘𝐴) ∈ On | |
2 | eleq1 2821 | . . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 236 | . . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | eloni 6183 | . . . . . . . 8 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → Ord 𝐴) |
6 | ordom 7611 | . . . . . . 7 ⊢ Ord ω | |
7 | ordtri2or 6268 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) | |
8 | 5, 6, 7 | sylancl 589 | . . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) |
9 | 8 | ord 863 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴)) |
10 | isinfcard 9595 | . . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ) | |
11 | 10 | biimpi 219 | . . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ) |
12 | 11 | expcom 417 | . . . . 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 862 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
15 | cardnn 9468 | . . . 4 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
16 | 10 | bicomi 227 | . . . . 5 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴)) |
17 | 16 | simprbi 500 | . . . 4 ⊢ (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴) |
18 | 15, 17 | jaoi 856 | . . 3 ⊢ ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴) |
19 | 14, 18 | impbii 212 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
20 | elun 4040 | . 2 ⊢ (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) | |
21 | 19, 20 | bitr4i 281 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∪ cun 3842 ⊆ wss 3844 ran crn 5527 Ord word 6172 Oncon0 6173 ‘cfv 6340 ωcom 7602 cardccrd 9440 ℵcale 9441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-om 7603 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-oi 9050 df-har 9097 df-card 9444 df-aleph 9445 |
This theorem is referenced by: cardnum 9597 carduniima 9599 cardinfima 9600 cfpwsdom 10087 gch2 10178 |
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