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Mirrors > Home > MPE Home > Th. List > nnsdomel | Structured version Visualization version GIF version |
Description: Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
nnsdomel | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardnn 9188 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
2 | cardnn 9188 | . . 3 ⊢ (𝐵 ∈ ω → (card‘𝐵) = 𝐵) | |
3 | eleq12 2855 | . . 3 ⊢ (((card‘𝐴) = 𝐴 ∧ (card‘𝐵) = 𝐵) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) | |
4 | 1, 2, 3 | syl2an 586 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) |
5 | nnon 7404 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
6 | onenon 9174 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ dom card) |
8 | nnon 7404 | . . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
9 | onenon 9174 | . . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom card) |
11 | cardsdom2 9213 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) | |
12 | 7, 10, 11 | syl2an 586 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) |
13 | 4, 12 | bitr3d 273 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 class class class wbr 4930 dom cdm 5408 Oncon0 6031 ‘cfv 6190 ωcom 7398 ≺ csdm 8307 cardccrd 9160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-om 7399 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-card 9164 |
This theorem is referenced by: fin23lem27 9550 |
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