|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > iscard | Structured version Visualization version GIF version | ||
| Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) | 
| Ref | Expression | 
|---|---|
| iscard | ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cardon 9985 | . . 3 ⊢ (card‘𝐴) ∈ On | |
| 2 | eleq1 2828 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
| 3 | 1, 2 | mpbii 233 | . 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) | 
| 4 | cardonle 9998 | . . . 4 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) | |
| 5 | eqss 3998 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))) | |
| 6 | 5 | baibr 536 | . . . 4 ⊢ ((card‘𝐴) ⊆ 𝐴 → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴)) | 
| 7 | 4, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ (card‘𝐴) = 𝐴)) | 
| 8 | dfss3 3971 | . . . 4 ⊢ (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (card‘𝐴)) | |
| 9 | onelon 6408 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
| 10 | onenon 9990 | . . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
| 11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ dom card) | 
| 12 | cardsdomel 10015 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) | |
| 13 | 9, 11, 12 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) | 
| 14 | 13 | ralbidva 3175 | . . . 4 ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (card‘𝐴))) | 
| 15 | 8, 14 | bitr4id 290 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | 
| 16 | 7, 15 | bitr3d 281 | . 2 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | 
| 17 | 3, 16 | biadanii 821 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ⊆ wss 3950 class class class wbr 5142 dom cdm 5684 Oncon0 6383 ‘cfv 6560 ≺ csdm 8985 cardccrd 9976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-card 9980 | 
| This theorem is referenced by: cardprclem 10020 cardmin2 10040 infxpenlem 10054 alephsuc2 10121 cardmin 10605 alephreg 10623 pwcfsdom 10624 winalim2 10737 gchina 10740 inar1 10816 r1tskina 10823 gruina 10859 iscard5 43554 | 
| Copyright terms: Public domain | W3C validator |