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Mirrors > Home > MPE Home > Th. List > brdom | Structured version Visualization version GIF version |
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
bren.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brdom | ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | brdomg 8616 | . 2 ⊢ (𝐵 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 class class class wbr 5039 –1-1→wf1 6355 ≼ cdom 8602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-dm 5546 df-rn 5547 df-fn 6361 df-f 6362 df-f1 6363 df-dom 8606 |
This theorem is referenced by: domen 8619 domtr 8659 sbthlem10 8743 1sdom 8857 ac10ct 9613 domtriomlem 10021 2ndcdisj 22307 birthdaylem3 25790 |
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