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Mirrors > Home > MPE Home > Th. List > sorpss | Structured version Visualization version GIF version |
Description: Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
sorpss | ⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | porpss 7453 | . . 3 ⊢ [⊊] Po 𝐴 | |
2 | 1 | biantrur 533 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥) ↔ ( [⊊] Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))) |
3 | sspsstri 4079 | . . . 4 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥)) | |
4 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 4 | brrpss 7452 | . . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦) |
6 | biid 263 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
7 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7 | brrpss 7452 | . . . . 5 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥) |
9 | 5, 6, 8 | 3orbi123i 1152 | . . . 4 ⊢ ((𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥)) |
10 | 3, 9 | bitr4i 280 | . . 3 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)) |
11 | 10 | 2ralbii 3166 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)) |
12 | df-so 5475 | . 2 ⊢ ( [⊊] Or 𝐴 ↔ ( [⊊] Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))) | |
13 | 2, 11, 12 | 3bitr4ri 306 | 1 ⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 ∀wral 3138 ⊆ wss 3936 ⊊ wpss 3937 class class class wbr 5066 Po wpo 5472 Or wor 5473 [⊊] 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-3or 1084 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-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-rpss 7449 |
This theorem is referenced by: sorpsscmpl 7460 enfin2i 9743 fin1a2lem13 9834 |
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