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Mirrors > Home > MPE Home > Th. List > sdomsdomcardi | Structured version Visualization version GIF version |
Description: A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.) |
Ref | Expression |
---|---|
sdomsdomcardi | ⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdom0 8380 | . . . . 5 ⊢ ¬ 𝐴 ≺ ∅ | |
2 | ndmfv 6476 | . . . . . 6 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅) | |
3 | 2 | breq2d 4898 | . . . . 5 ⊢ (¬ 𝐵 ∈ dom card → (𝐴 ≺ (card‘𝐵) ↔ 𝐴 ≺ ∅)) |
4 | 1, 3 | mtbiri 319 | . . . 4 ⊢ (¬ 𝐵 ∈ dom card → ¬ 𝐴 ≺ (card‘𝐵)) |
5 | 4 | con4i 114 | . . 3 ⊢ (𝐴 ≺ (card‘𝐵) → 𝐵 ∈ dom card) |
6 | cardid2 9112 | . . 3 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ≺ (card‘𝐵) → (card‘𝐵) ≈ 𝐵) |
8 | sdomentr 8382 | . 2 ⊢ ((𝐴 ≺ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≺ 𝐵) | |
9 | 7, 8 | mpdan 677 | 1 ⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ∅c0 4140 class class class wbr 4886 dom cdm 5355 ‘cfv 6135 ≈ cen 8238 ≺ csdm 8240 cardccrd 9094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-card 9098 |
This theorem is referenced by: sdomsdomcard 9717 |
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