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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > imadisjlnd | Structured version Visualization version GIF version |
Description: Natural deduction form of one negated side of imadisj 6076. (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 6076 | . . . 4 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) | |
3 | 2 | biimpi 215 | . . 3 ⊢ ((𝐴 “ 𝐵) = ∅ → (dom 𝐴 ∩ 𝐵) = ∅) |
4 | 3 | necon3i 2974 | . 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 1542 ≠ wne 2941 ∩ cin 3946 ∅c0 4321 dom cdm 5675 “ cima 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 |
This theorem is referenced by: wnefimgd 42846 |
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