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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 3968) 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 38501 | . . . . 5 ⊢ Rel S | |
| 2 | 1 | brrelex1i 5741 | . . . 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 5323 | . . . 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 4009 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦)) | |
| 11 | sseq2 4010 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵)) | |
| 12 | df-ssr 38499 | . . 3 ⊢ S = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} | |
| 13 | 10, 11, 12 | brabg 5544 | . 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 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 S cssr 38185 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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 df-ssr 38499 |
| This theorem is referenced by: brssrid 38503 brssrres 38505 brcnvssr 38507 extssr 38510 dfrefrels2 38514 dfsymrels2 38546 dftrrels2 38576 |
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