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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 5058 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵)) | |
2 | elex 3212 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
3 | 2 | ad2antll 765 | . . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ V) |
4 | brrpssg 6939 | . . . . 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 252 | . . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ 𝐵 = 𝐶)) | |
7 | elex 3212 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
8 | 7 | ad2antrl 764 | . . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ V) |
9 | brrpssg 6939 | . . . . 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 1398 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵) ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵))) |
12 | 1, 11 | mpbid 222 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵)) |
13 | sspsstri 3709 | . 2 ⊢ ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵)) | |
14 | 12, 13 | sylibr 224 | 1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ⊊ wpss 3575 class class class wbr 4653 Or wor 5034 [⊊] crpss 6936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-so 5036 df-xp 5120 df-rel 5121 df-rpss 6937 |
This theorem is referenced by: sorpssun 6944 sorpssin 6945 sorpssuni 6946 sorpssint 6947 sorpsscmpl 6948 enfin2i 9143 fin1a2lem9 9230 fin1a2lem10 9231 fin1a2lem11 9232 fin1a2lem13 9234 |
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