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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 9968 | . . . . . . . . 9 ⊢ (card‘𝐴) ∈ On | |
2 | eleq1 2817 | . . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 232 | . . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | eloni 6379 | . . . . . . . 8 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → Ord 𝐴) |
6 | ordom 7880 | . . . . . . 7 ⊢ Ord ω | |
7 | ordtri2or 6467 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) | |
8 | 5, 6, 7 | sylancl 585 | . . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) |
9 | 8 | ord 863 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴)) |
10 | isinfcard 10116 | . . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ) | |
11 | 10 | biimpi 215 | . . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ) |
12 | 11 | expcom 413 | . . . . 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 9987 | . . . 4 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
16 | 10 | bicomi 223 | . . . . 5 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴)) |
17 | 16 | simprbi 496 | . . . 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 208 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
20 | elun 4147 | . 2 ⊢ (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) | |
21 | 19, 20 | bitr4i 278 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 ran crn 5679 Ord word 6368 Oncon0 6369 ‘cfv 6548 ωcom 7870 cardccrd 9959 ℵcale 9960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9534 df-har 9581 df-card 9963 df-aleph 9964 |
This theorem is referenced by: cardnum 10118 carduniima 10120 cardinfima 10121 cfpwsdom 10608 gch2 10699 |
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