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Mirrors > Home > MPE Home > Th. List > domrefg | Structured version Visualization version GIF version |
Description: Dominance is reflexive. (Contributed by NM, 18-Jun-1998.) |
Ref | Expression |
---|---|
domrefg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrefg 8743 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
2 | endom 8738 | . 2 ⊢ (𝐴 ≈ 𝐴 → 𝐴 ≼ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2109 class class class wbr 5078 ≈ cen 8704 ≼ cdom 8705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-en 8708 df-dom 8709 |
This theorem is referenced by: cardprclem 9721 indcardi 9781 djudom1 9922 infdif 9949 alephexp2 10321 pwcfsdom 10323 alephom 10325 iunctb2 35553 sn1dom 41095 fvconstdomi 46139 indthincALT 46286 |
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