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Mirrors > Home > MPE Home > Th. List > cardval2 | Structured version Visualization version GIF version |
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10125. This theorem could be used to give a simpler definition of card in place of df-card 9520. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.) |
Ref | Expression |
---|---|
cardval2 | ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9525 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
2 | 1 | oneli 6299 | . . . . 5 ⊢ (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On) |
3 | 2 | pm4.71ri 564 | . . . 4 ⊢ (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))) |
4 | cardsdomel 9555 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) | |
5 | 4 | ancoms 462 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) |
6 | 5 | pm5.32da 582 | . . . 4 ⊢ (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥 ≺ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))) |
7 | 3, 6 | bitr4id 293 | . . 3 ⊢ (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴))) |
8 | 7 | abbi2dv 2867 | . 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)}) |
9 | df-rab 3060 | . 2 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)} | |
10 | 8, 9 | eqtr4di 2789 | 1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2714 {crab 3055 class class class wbr 5039 dom cdm 5536 Oncon0 6191 ‘cfv 6358 ≺ csdm 8603 cardccrd 9516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-card 9520 |
This theorem is referenced by: ondomon 10142 alephsuc3 10159 |
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