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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 8993 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
2 | cardnn 8993 | . . 3 ⊢ (𝐵 ∈ ω → (card‘𝐵) = 𝐵) | |
3 | eleq12 2840 | . . 3 ⊢ (((card‘𝐴) = 𝐴 ∧ (card‘𝐵) = 𝐵) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) | |
4 | 1, 2, 3 | syl2an 583 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) |
5 | nnon 7222 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
6 | onenon 8979 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ dom card) |
8 | nnon 7222 | . . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
9 | onenon 8979 | . . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom card) |
11 | cardsdom2 9018 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) | |
12 | 7, 10, 11 | syl2an 583 | . 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 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4787 dom cdm 5250 Oncon0 5865 ‘cfv 6030 ωcom 7216 ≺ csdm 8112 cardccrd 8965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-om 7217 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8969 |
This theorem is referenced by: fin23lem27 9356 |
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