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Mirrors > Home > MPE Home > Th. List > Mathboxes > brssr | Structured version Visualization version GIF version |
Description: The subset relation and subclass relationship (df-ss 3979) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.) |
Ref | Expression |
---|---|
brssr | ⊢ (𝐵 ∈ 𝑉 → (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relssr 38481 | . . . . 5 ⊢ Rel S | |
2 | 1 | brrelex1i 5744 | . . . 4 ⊢ (𝐴 S 𝐵 → 𝐴 ∈ V) |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 S 𝐵) → 𝐴 ∈ V) |
4 | simpl 482 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 S 𝐵) → 𝐵 ∈ 𝑉) | |
5 | 3, 4 | jca 511 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 S 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
6 | ssexg 5328 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
7 | simpr 484 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
8 | 6, 7 | jca 511 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
9 | 8 | ancoms 458 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝑉)) |
10 | sseq1 4020 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦)) | |
11 | sseq2 4021 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵)) | |
12 | df-ssr 38479 | . . 3 ⊢ S = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} | |
13 | 10, 11, 12 | brabg 5548 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
14 | 5, 9, 13 | pm5.21nd 802 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 class class class wbr 5147 S cssr 38164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-ssr 38479 |
This theorem is referenced by: brssrid 38483 brssrres 38485 brcnvssr 38487 extssr 38490 dfrefrels2 38494 dfsymrels2 38526 dftrrels2 38556 |
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