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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 9090 | . . . . . . . . 9 ⊢ (card‘𝐴) ∈ On | |
2 | eleq1 2894 | . . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 225 | . . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | eloni 5977 | . . . . . . . 8 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → Ord 𝐴) |
6 | ordom 7340 | . . . . . . 7 ⊢ Ord ω | |
7 | ordtri2or 6062 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) | |
8 | 5, 6, 7 | sylancl 580 | . . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) |
9 | 8 | ord 895 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴)) |
10 | isinfcard 9235 | . . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ) | |
11 | 10 | biimpi 208 | . . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ) |
12 | 11 | expcom 404 | . . . . 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 894 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
15 | cardnn 9109 | . . . 4 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
16 | 10 | bicomi 216 | . . . . 5 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴)) |
17 | 16 | simprbi 492 | . . . 4 ⊢ (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴) |
18 | 15, 17 | jaoi 888 | . . 3 ⊢ ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴) |
19 | 14, 18 | impbii 201 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
20 | elun 3982 | . 2 ⊢ (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) | |
21 | 19, 20 | bitr4i 270 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ∪ cun 3796 ⊆ wss 3798 ran crn 5347 Ord word 5966 Oncon0 5967 ‘cfv 6127 ωcom 7331 cardccrd 9081 ℵcale 9082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-oi 8691 df-har 8739 df-card 9085 df-aleph 9086 |
This theorem is referenced by: cardnum 9237 carduniima 9239 cardinfima 9240 cfpwsdom 9728 gch2 9819 |
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