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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 7721 | . . 3 ⊢ [⊊] Po 𝐴 | |
2 | 1 | biantrur 529 | . 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 4103 | . . . 4 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥)) | |
4 | vex 3476 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 4 | brrpss 7720 | . . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦) |
6 | biid 260 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
7 | vex 3476 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7 | brrpss 7720 | . . . . 5 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥) |
9 | 5, 6, 8 | 3orbi123i 1154 | . . . 4 ⊢ ((𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥)) |
10 | 3, 9 | bitr4i 277 | . . 3 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)) |
11 | 10 | 2ralbii 3126 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)) |
12 | df-so 5590 | . 2 ⊢ ( [⊊] Or 𝐴 ↔ ( [⊊] Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))) | |
13 | 2, 11, 12 | 3bitr4ri 303 | 1 ⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∨ wo 843 ∨ w3o 1084 ∀wral 3059 ⊆ wss 3949 ⊊ wpss 3950 class class class wbr 5149 Po wpo 5587 Or wor 5588 [⊊] crpss 7716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-rpss 7717 |
This theorem is referenced by: sorpsscmpl 7728 enfin2i 10320 fin1a2lem13 10411 |
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