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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscard5 | Structured version Visualization version GIF version |
Description: Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.) |
Ref | Expression |
---|---|
iscard5 | ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscard 9556 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | |
2 | sdomnen 8635 | . . . . 5 ⊢ (𝑥 ≺ 𝐴 → ¬ 𝑥 ≈ 𝐴) | |
3 | onelss 6233 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
4 | ssdomg 8652 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴)) | |
5 | 3, 4 | syld 47 | . . . . . . 7 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ≼ 𝐴)) |
6 | 5 | imp 410 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ≼ 𝐴) |
7 | brsdom 8629 | . . . . . . . 8 ⊢ (𝑥 ≺ 𝐴 ↔ (𝑥 ≼ 𝐴 ∧ ¬ 𝑥 ≈ 𝐴)) | |
8 | 7 | biimpri 231 | . . . . . . 7 ⊢ ((𝑥 ≼ 𝐴 ∧ ¬ 𝑥 ≈ 𝐴) → 𝑥 ≺ 𝐴) |
9 | 8 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → ((𝑥 ≼ 𝐴 ∧ ¬ 𝑥 ≈ 𝐴) → 𝑥 ≺ 𝐴)) |
10 | 6, 9 | mpand 695 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ≈ 𝐴 → 𝑥 ≺ 𝐴)) |
11 | 2, 10 | impbid2 229 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (𝑥 ≺ 𝐴 ↔ ¬ 𝑥 ≈ 𝐴)) |
12 | 11 | ralbidva 3107 | . . 3 ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
13 | 12 | pm5.32i 578 | . 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴) ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
14 | 1, 13 | bitri 278 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 class class class wbr 5039 Oncon0 6191 ‘cfv 6358 ≈ cen 8601 ≼ cdom 8602 ≺ 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: elrncard 40768 |
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