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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 8979 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
2 | cardnn 8979 | . . 3 ⊢ (𝐵 ∈ ω → (card‘𝐵) = 𝐵) | |
3 | eleq12 2829 | . . 3 ⊢ (((card‘𝐴) = 𝐴 ∧ (card‘𝐵) = 𝐵) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) | |
4 | 1, 2, 3 | syl2an 495 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) |
5 | nnon 7236 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
6 | onenon 8965 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ dom card) |
8 | nnon 7236 | . . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
9 | onenon 8965 | . . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom card) |
11 | cardsdom2 9004 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) | |
12 | 7, 10, 11 | syl2an 495 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) |
13 | 4, 12 | bitr3d 270 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 dom cdm 5266 Oncon0 5884 ‘cfv 6049 ωcom 7230 ≺ csdm 8120 cardccrd 8951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-om 7231 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 |
This theorem is referenced by: fin23lem27 9342 |
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