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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-brrelex12ALT | Structured version Visualization version GIF version | ||
| Description: Two classes related by a binary relation are both sets. Alternate proof of brrelex12 5737. (Contributed by BJ, 14-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-brrelex12ALT | ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0nelrel0 5745 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) | |
| 2 | jcn 162 | . . . 4 ⊢ (𝐴𝑅𝐵 → (¬ ∅ ∈ 𝑅 → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅))) | |
| 3 | 2 | impcom 407 | . . 3 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅)) | 
| 4 | opprc 4896 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
| 5 | df-br 5144 | . . . . . 6 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 6 | 5 | biimpi 216 | . . . . 5 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ 𝑅) | 
| 7 | eleq1 2829 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = ∅ → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ ∅ ∈ 𝑅)) | |
| 8 | 6, 7 | imbitrid 244 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = ∅ → (𝐴𝑅𝐵 → ∅ ∈ 𝑅)) | 
| 9 | 4, 8 | syl 17 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 → ∅ ∈ 𝑅)) | 
| 10 | 3, 9 | nsyl2 141 | . 2 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| 11 | 1, 10 | sylan 580 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 class class class wbr 5143 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: (None) | 
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