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Mirrors > Home > MPE Home > Th. List > neleq12d | Structured version Visualization version GIF version |
Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
Ref | Expression |
---|---|
neleq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
neleq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
neleq12d | ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neleq12d.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | neleq12d.2 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | 1, 2 | eleq12d 2828 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
4 | 3 | notbid 318 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐷)) |
5 | df-nel 3047 | . 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶) | |
6 | df-nel 3047 | . 2 ⊢ (𝐵 ∉ 𝐷 ↔ ¬ 𝐵 ∈ 𝐷) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∉ wnel 3046 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-clel 2811 df-nel 3047 |
This theorem is referenced by: neleq1 3051 neleq2 3052 uhgrspan1 28293 nbgrnself 28349 nbgrnself2 28350 finsumvtxdg2size 28540 fsetsnprcnex 45375 |
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