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Mirrors > Home > MPE Home > Th. List > sdomirr | Structured version Visualization version GIF version |
Description: Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.) |
Ref | Expression |
---|---|
sdomirr | ⊢ ¬ 𝐴 ≺ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomnen 8252 | . . 3 ⊢ (𝐴 ≺ 𝐴 → ¬ 𝐴 ≈ 𝐴) | |
2 | enrefg 8255 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | nsyl3 136 | . 2 ⊢ (𝐴 ∈ V → ¬ 𝐴 ≺ 𝐴) |
4 | relsdom 8230 | . . . 4 ⊢ Rel ≺ | |
5 | 4 | brrelex1i 5394 | . . 3 ⊢ (𝐴 ≺ 𝐴 → 𝐴 ∈ V) |
6 | 5 | con3i 152 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ≺ 𝐴) |
7 | 3, 6 | pm2.61i 177 | 1 ⊢ ¬ 𝐴 ≺ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2166 Vcvv 3415 class class class wbr 4874 ≈ cen 8220 ≺ csdm 8222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-en 8224 df-dom 8225 df-sdom 8226 |
This theorem is referenced by: sdomn2lp 8369 2pwuninel 8385 2pwne 8386 r111 8916 alephval2 9710 alephom 9723 csdfil 22069 |
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