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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 8506 | . . 3 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≺ ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
3 | df-br 5060 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
4 | df-br 5060 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
5 | df-br 5060 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 5 | notbii 322 | . . . 4 ⊢ (¬ 𝐴 ≈ 𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ≈ ) |
7 | 4, 6 | anbi12i 628 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) |
8 | eldif 3946 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
9 | 7, 8 | bitr4i 280 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
10 | 2, 3, 9 | 3bitr4i 305 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∖ cdif 3933 〈cop 4567 class class class wbr 5059 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-br 5060 df-sdom 8506 |
This theorem is referenced by: sdomdom 8531 sdomnen 8532 0sdomg 8640 sdomdomtr 8644 domsdomtr 8646 domtriord 8657 canth2 8664 php2 8696 php3 8697 nnsdomo 8707 nnsdomg 8771 card2inf 9013 cardsdomelir 9396 cardsdom2 9411 fidomtri2 9417 cardmin2 9421 alephordi 9494 alephord 9495 isfin4p1 9731 isfin5-2 9807 canthnum 10065 canthwe 10067 canthp1 10070 gchdjuidm 10084 gchxpidm 10085 gchhar 10095 axgroth6 10244 hashsdom 13736 ruc 15590 iscard5 39894 |
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