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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 9394 | . . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴) | |
2 | cardnn 9394 | . . 3 ⊢ (𝐵 ∈ ω → (card‘𝐵) = 𝐵) | |
3 | eleq12 2904 | . . 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 7588 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
6 | onenon 9380 | . . . 4 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ dom card) |
8 | nnon 7588 | . . . 4 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
9 | onenon 9380 | . . . 4 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom card) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom card) |
11 | cardsdom2 9419 | . . 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 283 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 dom cdm 5557 Oncon0 6193 ‘cfv 6357 ωcom 7582 ≺ csdm 8510 cardccrd 9366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 |
This theorem is referenced by: fin23lem27 9752 |
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