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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > imadisjld | Structured version Visualization version GIF version |
Description: Natural dduction form of one side of imadisj 6030. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
imadisjld.1 | ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅) |
Ref | Expression |
---|---|
imadisjld | ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadisjld.1 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅) | |
2 | imadisj 6030 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∩ cin 3907 ∅c0 4280 dom cdm 5631 “ cima 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 |
This theorem is referenced by: (None) |
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