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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 7844 | . . 3 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≺ ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
3 | df-br 4584 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
4 | df-br 4584 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
5 | df-br 4584 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 5 | notbii 309 | . . . 4 ⊢ (¬ 𝐴 ≈ 𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ≈ ) |
7 | 4, 6 | anbi12i 729 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) |
8 | eldif 3550 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≼ ∧ ¬ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
9 | 7, 8 | bitr4i 266 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≼ ∖ ≈ )) |
10 | 2, 3, 9 | 3bitr4i 291 | 1 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∖ cdif 3537 〈cop 4131 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-br 4584 df-sdom 7844 |
This theorem is referenced by: sdomdom 7869 sdomnen 7870 0sdomg 7974 sdomdomtr 7978 domsdomtr 7980 domtriord 7991 canth2 7998 php2 8030 php3 8031 nnsdomo 8040 nnsdomg 8104 card2inf 8343 cardsdomelir 8682 cardsdom2 8697 fidomtri2 8703 cardmin2 8707 alephordi 8780 alephord 8781 isfin4-3 9020 isfin5-2 9096 canthnum 9350 canthwe 9352 canthp1 9355 gchcdaidm 9369 gchxpidm 9370 gchhar 9380 axgroth6 9529 hashsdom 13031 ruc 14811 |
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