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 8653 | . . . . . . . . 9 ⊢ (card‘𝐴) ∈ On | |
2 | eleq1 2676 | . . . . . . . . 9 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 222 | . . . . . . . 8 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | eloni 5650 | . . . . . . . 8 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ ((card‘𝐴) = 𝐴 → Ord 𝐴) |
6 | ordom 6966 | . . . . . . 7 ⊢ Ord ω | |
7 | ordtri2or 5739 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) | |
8 | 5, 6, 7 | sylancl 693 | . . . . . 6 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ ω ⊆ 𝐴)) |
9 | 8 | ord 391 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → ω ⊆ 𝐴)) |
10 | isinfcard 8798 | . . . . . . 7 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ) | |
11 | 10 | biimpi 205 | . . . . . 6 ⊢ ((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 ∈ ran ℵ) |
12 | 11 | expcom 450 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 → 𝐴 ∈ ran ℵ)) |
13 | 9, 12 | syld 46 | . . . 4 ⊢ ((card‘𝐴) = 𝐴 → (¬ 𝐴 ∈ ω → 𝐴 ∈ ran ℵ)) |
14 | 13 | orrd 392 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
15 | cardnn 8672 | . . . 4 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
16 | 10 | bicomi 213 | . . . . 5 ⊢ (𝐴 ∈ ran ℵ ↔ (ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴)) |
17 | 16 | simprbi 479 | . . . 4 ⊢ (𝐴 ∈ ran ℵ → (card‘𝐴) = 𝐴) |
18 | 15, 17 | jaoi 393 | . . 3 ⊢ ((𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ) → (card‘𝐴) = 𝐴) |
19 | 14, 18 | impbii 198 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) |
20 | elun 3715 | . 2 ⊢ (𝐴 ∈ (ω ∪ ran ℵ) ↔ (𝐴 ∈ ω ∨ 𝐴 ∈ ran ℵ)) | |
21 | 19, 20 | bitr4i 266 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ (ω ∪ ran ℵ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 ran crn 5039 Ord word 5639 Oncon0 5640 ‘cfv 5804 ωcom 6957 cardccrd 8644 ℵcale 8645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-har 8346 df-card 8648 df-aleph 8649 |
This theorem is referenced by: cardnum 8800 carduniima 8802 cardinfima 8803 cfpwsdom 9285 gch2 9376 |
Copyright terms: Public domain | W3C validator |