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Mirrors > Home > MPE Home > Th. List > sucdom | Structured version Visualization version GIF version |
Description: Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
sucdom | ⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucdom2 8711 | . 2 ⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) | |
2 | php4 8701 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) | |
3 | sdomdomtr 8647 | . . . 4 ⊢ ((𝐴 ≺ suc 𝐴 ∧ suc 𝐴 ≼ 𝐵) → 𝐴 ≺ 𝐵) | |
4 | 3 | ex 415 | . . 3 ⊢ (𝐴 ≺ suc 𝐴 → (suc 𝐴 ≼ 𝐵 → 𝐴 ≺ 𝐵)) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → (suc 𝐴 ≼ 𝐵 → 𝐴 ≺ 𝐵)) |
6 | 1, 5 | impbid2 228 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2113 class class class wbr 5063 suc csuc 6190 ωcom 7577 ≼ cdom 8504 ≺ csdm 8505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-om 7578 df-1o 8099 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 |
This theorem is referenced by: 0sdom1dom 8713 1sdom 8718 isnzr2 20032 harsucnn 39977 |
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