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Mirrors > Home > MPE Home > Th. List > brsdom | Structured version Visualization version GIF version |
Description: Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴." Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.) |
Ref | Expression |
---|---|
brsdom | ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sdom 7958 | . . 3 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≺ ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
3 | df-br 4654 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
4 | df-br 4654 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
5 | df-br 4654 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 5 | notbii 310 | . . . 4 ⊢ (¬ 𝐴 ≈ 𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ≈ ) |
7 | 4, 6 | anbi12i 733 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) |
8 | eldif 3584 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
9 | 7, 8 | bitr4i 267 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
10 | 2, 3, 9 | 3bitr4i 292 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ∖ cdif 3571 〈cop 4183 class class class wbr 4653 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-br 4654 df-sdom 7958 |
This theorem is referenced by: sdomdom 7983 sdomnen 7984 0sdomg 8089 sdomdomtr 8093 domsdomtr 8095 domtriord 8106 canth2 8113 php2 8145 php3 8146 nnsdomo 8155 nnsdomg 8219 card2inf 8460 cardsdomelir 8799 cardsdom2 8814 fidomtri2 8820 cardmin2 8824 alephordi 8897 alephord 8898 isfin4-3 9137 isfin5-2 9213 canthnum 9471 canthwe 9473 canthp1 9476 gchcdaidm 9490 gchxpidm 9491 gchhar 9501 axgroth6 9650 hashsdom 13170 ruc 14972 |
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