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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 8760 | . 2 ⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) | |
2 | php4 8838 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) | |
3 | sdomdomtr 8784 | . . . 4 ⊢ ((𝐴 ≺ suc 𝐴 ∧ suc 𝐴 ≼ 𝐵) → 𝐴 ≺ 𝐵) | |
4 | 3 | ex 416 | . . 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 229 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2110 class class class wbr 5058 suc csuc 6220 ωcom 7649 ≼ cdom 8629 ≺ csdm 8630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-br 5059 df-opab 5121 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-om 7650 df-er 8396 df-en 8632 df-dom 8633 df-sdom 8634 |
This theorem is referenced by: 0sdom1dom 8881 1sdom 8886 harsucnn 9619 isnzr2 20306 |
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