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Mirrors > Home > MPE Home > Th. List > sorpssi | Structured version Visualization version GIF version |
Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
sorpssi | ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | solin 5612 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵)) | |
2 | elex 3492 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
3 | 2 | ad2antll 727 | . . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ V) |
4 | brrpssg 7711 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐵 [⊊] 𝐶 ↔ 𝐵 ⊊ 𝐶)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 [⊊] 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
6 | biidd 261 | . . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ 𝐵 = 𝐶)) | |
7 | elex 3492 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
8 | 7 | ad2antrl 726 | . . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ V) |
9 | brrpssg 7711 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐶 [⊊] 𝐵 ↔ 𝐶 ⊊ 𝐵)) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 [⊊] 𝐵 ↔ 𝐶 ⊊ 𝐵)) |
11 | 5, 6, 10 | 3orbi123d 1435 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵) ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵))) |
12 | 1, 11 | mpbid 231 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵)) |
13 | sspsstri 4101 | . 2 ⊢ ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵)) | |
14 | 12, 13 | sylibr 233 | 1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 ⊊ wpss 3948 class class class wbr 5147 Or wor 5586 [⊊] crpss 7708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-so 5588 df-xp 5681 df-rel 5682 df-rpss 7709 |
This theorem is referenced by: sorpssun 7716 sorpssin 7717 sorpssuni 7718 sorpssint 7719 sorpsscmpl 7720 enfin2i 10312 fin1a2lem9 10399 fin1a2lem10 10400 fin1a2lem11 10401 fin1a2lem13 10403 ssmxidllem 32577 |
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