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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 9644 and serves as an example in the context of Godel codes, see elnanelprv 35414. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elnanel | ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2lp 9644 | . 2 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | |
2 | df-nan 1489 | . 2 ⊢ ((𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
3 | 1, 2 | mpbir 231 | 1 ⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ⊼ wnan 1488 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-nan 1489 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-fr 5641 |
This theorem is referenced by: elnanelprv 35414 |
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