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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 5700. (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 5700 | . . . 4 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) | |
3 | 2 | biimpi 208 | . . 3 ⊢ ((𝐴 “ 𝐵) = ∅ → (dom 𝐴 ∩ 𝐵) = ∅) |
4 | 3 | necon3i 3002 | . 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 1653 ≠ wne 2970 ∩ cin 3767 ∅c0 4114 dom cdm 5311 “ cima 5314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pr 5096 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3386 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-sn 4368 df-pr 4370 df-op 4374 df-br 4843 df-opab 4905 df-xp 5317 df-cnv 5319 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 |
This theorem is referenced by: (None) |
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