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Mirrors > Home > MPE Home > Th. List > imadisjlnd | Structured version Visualization version GIF version |
Description: Deduction form of one negated side of imadisj 6081. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
imadisjlnd.1 | ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅) |
Ref | Expression |
---|---|
imadisjlnd | ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadisjlnd.1 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅) | |
2 | imadisj 6081 | . . . 4 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) | |
3 | 2 | biimpi 215 | . . 3 ⊢ ((𝐴 “ 𝐵) = ∅ → (dom 𝐴 ∩ 𝐵) = ∅) |
4 | 3 | necon3i 2963 | . 2 ⊢ ((dom 𝐴 ∩ 𝐵) ≠ ∅ → (𝐴 “ 𝐵) ≠ ∅) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ≠ wne 2930 ∩ cin 3947 ∅c0 4324 dom cdm 5674 “ cima 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3466 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5146 df-opab 5208 df-xp 5680 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 |
This theorem is referenced by: wnefimgd 43864 grimuhgr 47492 |
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