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| 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 9019 | . . 3 ⊢ ≼ = ( ≺ ∪ ≈ ) | |
| 2 | 1 | eleq2i 2832 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≼ ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) | 
| 3 | df-br 5143 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
| 4 | df-br 5143 | . . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
| 5 | df-br 5143 | . . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
| 6 | 4, 5 | orbi12i 914 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) | 
| 7 | elun 4152 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
| 8 | 6, 7 | bitr4i 278 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) | 
| 9 | 2, 3, 8 | 3bitr4i 303 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 ∈ wcel 2107 ∪ cun 3948 〈cop 4631 class class class wbr 5142 ≈ cen 8983 ≼ cdom 8984 ≺ csdm 8985 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-f1o 6567 df-en 8987 df-dom 8988 df-sdom 8989 | 
| This theorem is referenced by: bren2 9024 domnsym 9140 domnsymfi 9241 modom 9281 carddom2 10018 axcc4dom 10482 entric 10598 entri2 10599 gchor 10668 frgpcyg 21593 iunmbl2 25593 dyadmbl 25636 padct 32732 volmeas 34233 ovoliunnfl 37670 ctbnfien 42834 | 
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