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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 3501 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
| 2 | relrpss 7744 | . . . 4 ⊢ Rel [⊊] | |
| 3 | 2 | brrelex1i 5741 | . . 3 ⊢ (𝐴 [⊊] 𝐵 → 𝐴 ∈ V) |
| 4 | 1, 3 | anim12i 613 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 [⊊] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
| 5 | 1 | adantr 480 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐵 ∈ V) |
| 6 | pssss 4098 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
| 7 | ssexg 5323 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
| 8 | 6, 1, 7 | syl2anr 597 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐴 ∈ V) |
| 9 | 5, 8 | jca 511 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
| 10 | psseq1 4090 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦)) | |
| 11 | psseq2 4091 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵)) | |
| 12 | df-rpss 7743 | . . . 4 ⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} | |
| 13 | 10, 11, 12 | brabg 5544 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
| 14 | 13 | ancoms 458 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
| 15 | 4, 9, 14 | pm5.21nd 802 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ⊊ wpss 3952 class class class wbr 5143 [⊊] crpss 7742 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 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-rpss 7743 |
| This theorem is referenced by: brrpss 7746 sorpssi 7749 |
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