Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfsdom2 | Structured version Visualization version GIF version |
Description: Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
dfsdom2 | ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sdom 8500 | . 2 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | sbthcl 8627 | . . 3 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) | |
3 | 2 | difeq2i 4093 | . 2 ⊢ ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) |
4 | difin 4235 | . 2 ⊢ ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) = ( ≼ ∖ ◡ ≼ ) | |
5 | 1, 3, 4 | 3eqtri 2845 | 1 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∖ cdif 3930 ∩ cin 3932 ◡ccnv 5547 ≈ cen 8494 ≼ cdom 8495 ≺ csdm 8496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 |
This theorem is referenced by: brsdom2 8629 |
Copyright terms: Public domain | W3C validator |