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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 7870 | . . 3 ⊢ (𝐴 ≺ 𝐴 → ¬ 𝐴 ≈ 𝐴) | |
2 | enrefg 7873 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | nsyl3 132 | . 2 ⊢ (𝐴 ∈ V → ¬ 𝐴 ≺ 𝐴) |
4 | relsdom 7848 | . . . 4 ⊢ Rel ≺ | |
5 | 4 | brrelexi 5082 | . . 3 ⊢ (𝐴 ≺ 𝐴 → 𝐴 ∈ V) |
6 | 5 | con3i 149 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ≺ 𝐴) |
7 | 3, 6 | pm2.61i 175 | 1 ⊢ ¬ 𝐴 ≺ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ≈ cen 7838 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: sdomn2lp 7984 2pwuninel 8000 2pwne 8001 r111 8521 alephval2 9273 alephom 9286 csdfil 21508 |
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