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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 3512 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | relrpss 7450 | . . . 4 ⊢ Rel [⊊] | |
3 | 2 | brrelex1i 5608 | . . 3 ⊢ (𝐴 [⊊] 𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | anim12i 614 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 [⊊] 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
5 | 1 | adantr 483 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐵 ∈ V) |
6 | pssss 4072 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
7 | ssexg 5227 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
8 | 6, 1, 7 | syl2anr 598 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → 𝐴 ∈ V) |
9 | 5, 8 | jca 514 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) |
10 | psseq1 4064 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦)) | |
11 | psseq2 4065 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵)) | |
12 | df-rpss 7449 | . . . 4 ⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} | |
13 | 10, 11, 12 | brabg 5426 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
14 | 13 | ancoms 461 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
15 | 4, 9, 14 | pm5.21nd 800 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ⊊ wpss 3937 class class class wbr 5066 [⊊] crpss 7448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-rpss 7449 |
This theorem is referenced by: brrpss 7452 sorpssi 7455 |
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