![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8495 | . . 3 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≺ ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
3 | df-br 5031 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
4 | df-br 5031 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
5 | df-br 5031 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 5 | notbii 323 | . . . 4 ⊢ (¬ 𝐴 ≈ 𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ≈ ) |
7 | 4, 6 | anbi12i 629 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) |
8 | eldif 3891 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
9 | 7, 8 | bitr4i 281 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
10 | 2, 3, 9 | 3bitr4i 306 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∖ cdif 3878 〈cop 4531 class class class wbr 5030 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-br 5031 df-sdom 8495 |
This theorem is referenced by: sdomdom 8520 sdomnen 8521 0sdomg 8630 sdomdomtr 8634 domsdomtr 8636 domtriord 8647 canth2 8654 php2 8686 php3 8687 nnsdomo 8698 nnsdomg 8761 card2inf 9003 cardsdomelir 9386 cardsdom2 9401 fidomtri2 9407 cardmin2 9412 alephordi 9485 alephord 9486 isfin4p1 9726 isfin5-2 9802 canthnum 10060 canthwe 10062 canthp1 10065 gchdjuidm 10079 gchxpidm 10080 gchhar 10090 axgroth6 10239 hashsdom 13738 ruc 15588 iscard5 40242 |
Copyright terms: Public domain | W3C validator |