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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elnelneqd | Structured version Visualization version GIF version |
Description: Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
elnelneqd.1 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
elnelneqd.2 | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
Ref | Expression |
---|---|
elnelneqd | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnelneqd.2 | . 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) | |
2 | elnelneqd.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
3 | 2 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐴) |
4 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
5 | 3, 4 | eleqtrd 2892 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐵) |
6 | 1, 5 | mtand 815 | 1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: mnuprdlem1 40980 mnuprdlem2 40981 |
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