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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 8535 | . . 3 ⊢ ≼ = ( ≺ ∪ ≈ ) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≼ ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) |
3 | df-br 5067 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
4 | df-br 5067 | . . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
5 | df-br 5067 | . . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 4, 5 | orbi12i 911 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) |
7 | elun 4125 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
8 | 6, 7 | bitr4i 280 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) |
9 | 2, 3, 8 | 3bitr4i 305 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∨ wo 843 ∈ wcel 2114 ∪ cun 3934 〈cop 4573 class class class wbr 5066 ≈ cen 8506 ≼ cdom 8507 ≺ csdm 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-f1o 6362 df-en 8510 df-dom 8511 df-sdom 8512 |
This theorem is referenced by: bren2 8540 domnsym 8643 modom 8719 carddom2 9406 axcc4dom 9863 entric 9979 entri2 9980 gchor 10049 frgpcyg 20720 iunmbl2 24158 dyadmbl 24201 padct 30455 volmeas 31490 ovoliunnfl 34949 ctbnfien 39435 |
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