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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 8770 | . . 3 ⊢ (card‘𝐴) ∈ On | |
2 | eleq1 2689 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → ((card‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 223 | . 2 ⊢ ((card‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | cardonle 8783 | . . . 4 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) | |
5 | eqss 3618 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 ↔ ((card‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (card‘𝐴))) | |
6 | 5 | baibr 945 | . . . 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 | onelon 5748 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
9 | onenon 8775 | . . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
10 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ dom card) |
11 | cardsdomel 8800 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) | |
12 | 8, 10, 11 | syl2anc 693 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) |
13 | 12 | ralbidva 2985 | . . . 4 ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (card‘𝐴))) |
14 | dfss3 3592 | . . . 4 ⊢ (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (card‘𝐴)) | |
15 | 13, 14 | syl6rbbr 279 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (card‘𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) |
16 | 7, 15 | bitr3d 270 | . 2 ⊢ (𝐴 ∈ On → ((card‘𝐴) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) |
17 | 3, 16 | biadan2 674 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 Oncon0 5723 ‘cfv 5888 ≺ csdm 7954 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-card 8765 |
This theorem is referenced by: cardprclem 8805 cardmin2 8824 infxpenlem 8836 alephsuc2 8903 cardmin 9386 alephreg 9404 pwcfsdom 9405 winalim2 9518 gchina 9521 inar1 9597 r1tskina 9604 gruina 9640 |
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