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Mirrors > Home > MPE Home > Th. List > cardval3 | Structured version Visualization version GIF version |
Description: An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
cardval3 | ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3243 | . 2 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V) | |
2 | isnum2 8809 | . . . 4 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) | |
3 | rabn0 3991 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) | |
4 | intex 4850 | . . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) | |
5 | 2, 3, 4 | 3bitr2i 288 | . . 3 ⊢ (𝐴 ∈ dom card ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) |
6 | 5 | biimpi 206 | . 2 ⊢ (𝐴 ∈ dom card → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) |
7 | breq2 4689 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝑥 ≈ 𝐴)) | |
8 | 7 | rabbidv 3220 | . . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
9 | 8 | inteqd 4512 | . . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦} = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
10 | df-card 8803 | . . 3 ⊢ card = (𝑦 ∈ V ↦ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝑦}) | |
11 | 9, 10 | fvmptg 6319 | . 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
12 | 1, 6, 11 | syl2anc 694 | 1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 {crab 2945 Vcvv 3231 ∅c0 3948 ∩ cint 4507 class class class wbr 4685 dom cdm 5143 Oncon0 5761 ‘cfv 5926 ≈ cen 7994 cardccrd 8799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-ord 5764 df-on 5765 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-en 7998 df-card 8803 |
This theorem is referenced by: cardid2 8817 oncardval 8819 cardidm 8823 cardne 8829 cardval 9406 |
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