Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8574 | . . 3 ⊢ ≼ = ( ≺ ∪ ≈ ) | |
2 | 1 | eleq2i 2824 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ≼ ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) |
3 | df-br 5028 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≼ ) | |
4 | df-br 5028 | . . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≺ ) | |
5 | df-br 5028 | . . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≈ ) | |
6 | 4, 5 | orbi12i 914 | . . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) |
7 | elun 4037 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ ) ↔ (〈𝐴, 𝐵〉 ∈ ≺ ∨ 〈𝐴, 𝐵〉 ∈ ≈ )) | |
8 | 6, 7 | bitr4i 281 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ 〈𝐴, 𝐵〉 ∈ ( ≺ ∪ ≈ )) |
9 | 2, 3, 8 | 3bitr4i 306 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 846 ∈ wcel 2113 ∪ cun 3839 〈cop 4519 class class class wbr 5027 ≈ cen 8545 ≼ cdom 8546 ≺ csdm 8547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-xp 5525 df-rel 5526 df-f1o 6340 df-en 8549 df-dom 8550 df-sdom 8551 |
This theorem is referenced by: bren2 8579 domnsym 8686 modom 8791 carddom2 9472 axcc4dom 9934 entric 10050 entri2 10051 gchor 10120 frgpcyg 20385 iunmbl2 24302 dyadmbl 24345 padct 30621 volmeas 31761 ovoliunnfl 35431 ctbnfien 40196 |
Copyright terms: Public domain | W3C validator |