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Mirrors > Home > MPE Home > Th. List > brrpssg | Structured version Visualization version GIF version |
Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
brrpssg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3466 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | relrpss 7666 | . . . 4 ⊢ Rel [⊊] | |
3 | 2 | brrelex1i 5693 | . . 3 ⊢ (𝐴 [⊊] 𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | anim12i 614 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 [⊊] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
5 | 1 | adantr 482 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐵 ∈ V) |
6 | pssss 4060 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
7 | ssexg 5285 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
8 | 6, 1, 7 | syl2anr 598 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐴 ∈ V) |
9 | 5, 8 | jca 513 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
10 | psseq1 4052 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦)) | |
11 | psseq2 4053 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵)) | |
12 | df-rpss 7665 | . . . 4 ⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦} | |
13 | 10, 11, 12 | brabg 5501 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
14 | 13 | ancoms 460 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
15 | 4, 9, 14 | pm5.21nd 801 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3448 ⊆ wss 3915 ⊊ wpss 3916 class class class wbr 5110 [⊊] crpss 7664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-rpss 7665 |
This theorem is referenced by: brrpss 7668 sorpssi 7671 |
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