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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 9247. This theorem could be used to give us a simpler definition of card in place of df-card 8648. 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 | cardsdomel 8683 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) | |
2 | 1 | ancoms 468 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥 ≺ 𝐴 ↔ 𝑥 ∈ (card‘𝐴))) |
3 | 2 | pm5.32da 671 | . . . 4 ⊢ (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥 ≺ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))) |
4 | cardon 8653 | . . . . . 6 ⊢ (card‘𝐴) ∈ On | |
5 | 4 | oneli 5752 | . . . . 5 ⊢ (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On) |
6 | 5 | pm4.71ri 663 | . . . 4 ⊢ (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))) |
7 | 3, 6 | syl6rbbr 278 | . . 3 ⊢ (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴))) |
8 | 7 | abbi2dv 2729 | . 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)}) |
9 | df-rab 2905 | . 2 ⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ≺ 𝐴)} | |
10 | 8, 9 | syl6eqr 2662 | 1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 {crab 2900 class class class wbr 4583 dom cdm 5038 Oncon0 5640 ‘cfv 5804 ≺ csdm 7840 cardccrd 8644 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-rab 2905 df-v 3175 df-sbc 3403 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-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-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-ord 5643 df-on 5644 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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-card 8648 |
This theorem is referenced by: ondomon 9264 alephsuc3 9281 |
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