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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 3185 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | relrpss 6836 | . . . 4 ⊢ Rel [⊊] | |
3 | 2 | brrelexi 5082 | . . 3 ⊢ (𝐴 [⊊] 𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | anim12i 588 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 [⊊] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
5 | 1 | adantr 480 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐵 ∈ V) |
6 | pssss 3664 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
7 | ssexg 4732 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
8 | 6, 1, 7 | syl2anr 494 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐴 ∈ V) |
9 | 5, 8 | jca 553 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
10 | psseq1 3656 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦)) | |
11 | psseq2 3657 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵)) | |
12 | df-rpss 6835 | . . . 4 ⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} | |
13 | 10, 11, 12 | brabg 4919 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
14 | 13 | ancoms 468 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
15 | 4, 9, 14 | pm5.21nd 939 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ⊊ wpss 3541 class class class wbr 4583 [⊊] crpss 6834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-rpss 6835 |
This theorem is referenced by: brrpss 6838 sorpssi 6841 |
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