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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 9397 | . 2 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | |
2 | sdomnen 8531 | . . . . 5 ⊢ (𝑥 ≺ 𝐴 → ¬ 𝑥 ≈ 𝐴) | |
3 | onelss 6226 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
4 | ssdomg 8548 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴)) | |
5 | 3, 4 | syld 47 | . . . . . . 7 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ≼ 𝐴)) |
6 | 5 | imp 409 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ≼ 𝐴) |
7 | brsdom 8525 | . . . . . . . 8 ⊢ (𝑥 ≺ 𝐴 ↔ (𝑥 ≼ 𝐴 ∧ ¬ 𝑥 ≈ 𝐴)) | |
8 | 7 | biimpri 230 | . . . . . . 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 693 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ≈ 𝐴 → 𝑥 ≺ 𝐴)) |
11 | 2, 10 | impbid2 228 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ 𝐴) → (𝑥 ≺ 𝐴 ↔ ¬ 𝑥 ≈ 𝐴)) |
12 | 11 | ralbidva 3195 | . . 3 ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
13 | 12 | pm5.32i 577 | . 2 ⊢ ((𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴) ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
14 | 1, 13 | bitri 277 | 1 ⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ⊆ wss 3929 class class class wbr 5059 Oncon0 6184 ‘cfv 6348 ≈ cen 8499 ≼ cdom 8500 ≺ csdm 8501 cardccrd 9357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-card 9361 |
This theorem is referenced by: elrncard 39977 |
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