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Mirrors > Home > MPE Home > Th. List > elnanel | Structured version Visualization version GIF version |
Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9501 and serves as an example in the context of Godel codes, see elnanelprv 33827. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elnanel | ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2lp 9501 | . 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | |
2 | df-nan 1491 | . 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ⊼ wnan 1490 ∈ wcel 2107 |
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-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-reg 9487 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-nan 1491 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-eprel 5536 df-fr 5587 |
This theorem is referenced by: elnanelprv 33827 |
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