![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9833 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
2 | cardnn 9833 | . . 3 ⊢ (𝐵 ∈ ω → (card‘𝐵) = 𝐵) | |
3 | eleq12 2828 | . . 3 ⊢ (((card‘𝐴) = 𝐴 ∧ (card‘𝐵) = 𝐵) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ∈ 𝐵)) |
5 | nnon 7799 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
6 | onenon 9819 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ dom card) |
8 | nnon 7799 | . . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
9 | onenon 9819 | . . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom card) |
11 | cardsdom2 9858 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) | |
12 | 7, 10, 11 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) |
13 | 4, 12 | bitr3d 281 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5104 dom cdm 5631 Oncon0 6314 ‘cfv 6492 ωcom 7793 ≺ csdm 8816 cardccrd 9805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-om 7794 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-card 9809 |
This theorem is referenced by: fin23lem27 10198 |
Copyright terms: Public domain | W3C validator |