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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 5604. (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 5612 | . 2 ⊢ (Rel 𝑅 → ¬ ∅ ∈ 𝑅) | |
2 | jcn 164 | . . . 4 ⊢ (𝐴𝑅𝐵 → (¬ ∅ ∈ 𝑅 → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅))) | |
3 | 2 | impcom 410 | . . 3 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → ¬ (𝐴𝑅𝐵 → ∅ ∈ 𝑅)) |
4 | opprc 4826 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
5 | df-br 5067 | . . . . . 6 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
6 | 5 | biimpi 218 | . . . . 5 ⊢ (𝐴𝑅𝐵 → 〈𝐴, 𝐵〉 ∈ 𝑅) |
7 | eleq1 2900 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 = ∅ → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ ∅ ∈ 𝑅)) | |
8 | 6, 7 | syl5ib 246 | . . . 4 ⊢ (〈𝐴, 𝐵〉 = ∅ → (𝐴𝑅𝐵 → ∅ ∈ 𝑅)) |
9 | 4, 8 | syl 17 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 → ∅ ∈ 𝑅)) |
10 | 3, 9 | nsyl2 143 | . 2 ⊢ ((¬ ∅ ∈ 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 1, 10 | sylan 582 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 〈cop 4573 class class class wbr 5066 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: (None) |
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