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Mirrors > Home > MPE Home > Th. List > brdom2 | Structured version Visualization version GIF version |
Description: Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.) |
Ref | Expression |
---|---|
brdom2 | ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdom2 7867 | . . 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 | . . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 4, 5 | orbi12i 542 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) |
7 | elun 3715 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
8 | 6, 7 | bitr4i 266 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) |
9 | 2, 3, 8 | 3bitr4i 291 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∈ wcel 1977 ∪ cun 3538 〈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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-f1o 5811 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: bren2 7872 domnsym 7971 modom 8046 carddom2 8686 axcc4dom 9146 entric 9258 entri2 9259 gchor 9328 frgpcyg 19741 iunmbl2 23132 dyadmbl 23174 padct 28885 volmeas 29621 ovoliunnfl 32621 ctbnfien 36400 |
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