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Mirrors > Home > MPE Home > Th. List > porpss | Structured version Visualization version GIF version |
Description: Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Ref | Expression |
---|---|
porpss | ⊢ [⊊] Po 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssirr 4101 | . . . . 5 ⊢ ¬ 𝑥 ⊊ 𝑥 | |
2 | psstr 4105 | . . . . 5 ⊢ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧) | |
3 | vex 3479 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 3 | brrpss 7716 | . . . . . . 7 ⊢ (𝑥 [⊊] 𝑥 ↔ 𝑥 ⊊ 𝑥) |
5 | 4 | notbii 320 | . . . . . 6 ⊢ (¬ 𝑥 [⊊] 𝑥 ↔ ¬ 𝑥 ⊊ 𝑥) |
6 | vex 3479 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 6 | brrpss 7716 | . . . . . . . 8 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦) |
8 | vex 3479 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
9 | 8 | brrpss 7716 | . . . . . . . 8 ⊢ (𝑦 [⊊] 𝑧 ↔ 𝑦 ⊊ 𝑧) |
10 | 7, 9 | anbi12i 628 | . . . . . . 7 ⊢ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) ↔ (𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧)) |
11 | 8 | brrpss 7716 | . . . . . . 7 ⊢ (𝑥 [⊊] 𝑧 ↔ 𝑥 ⊊ 𝑧) |
12 | 10, 11 | imbi12i 351 | . . . . . 6 ⊢ (((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧) ↔ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)) |
13 | 5, 12 | anbi12i 628 | . . . . 5 ⊢ ((¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) ↔ (¬ 𝑥 ⊊ 𝑥 ∧ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧))) |
14 | 1, 2, 13 | mpbir2an 710 | . . . 4 ⊢ (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) |
15 | 14 | rgenw 3066 | . . 3 ⊢ ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) |
16 | 15 | rgen2w 3067 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) |
17 | df-po 5589 | . 2 ⊢ ( [⊊] Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))) | |
18 | 16, 17 | mpbir 230 | 1 ⊢ [⊊] Po 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∀wral 3062 ⊊ wpss 3950 class class class wbr 5149 Po wpo 5587 [⊊] crpss 7712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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-xp 5683 df-rel 5684 df-rpss 7713 |
This theorem is referenced by: sorpss 7718 fin23lem40 10346 isfin1-3 10381 zorng 10499 fin2so 36475 |
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