Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4074 | . . . . 5 ⊢ ¬ 𝑥 ⊊ 𝑥 | |
2 | psstr 4078 | . . . . 5 ⊢ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧) | |
3 | vex 3495 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 3 | brrpss 7441 | . . . . . . 7 ⊢ (𝑥 [⊊] 𝑥 ↔ 𝑥 ⊊ 𝑥) |
5 | 4 | notbii 321 | . . . . . 6 ⊢ (¬ 𝑥 [⊊] 𝑥 ↔ ¬ 𝑥 ⊊ 𝑥) |
6 | vex 3495 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
7 | 6 | brrpss 7441 | . . . . . . . 8 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦) |
8 | vex 3495 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
9 | 8 | brrpss 7441 | . . . . . . . 8 ⊢ (𝑦 [⊊] 𝑧 ↔ 𝑦 ⊊ 𝑧) |
10 | 7, 9 | anbi12i 626 | . . . . . . 7 ⊢ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) ↔ (𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧)) |
11 | 8 | brrpss 7441 | . . . . . . 7 ⊢ (𝑥 [⊊] 𝑧 ↔ 𝑥 ⊊ 𝑧) |
12 | 10, 11 | imbi12i 352 | . . . . . 6 ⊢ (((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧) ↔ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧)) |
13 | 5, 12 | anbi12i 626 | . . . . 5 ⊢ ((¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) ↔ (¬ 𝑥 ⊊ 𝑥 ∧ ((𝑥 ⊊ 𝑦 ∧ 𝑦 ⊊ 𝑧) → 𝑥 ⊊ 𝑧))) |
14 | 1, 2, 13 | mpbir2an 707 | . . . 4 ⊢ (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) |
15 | 14 | rgenw 3147 | . . 3 ⊢ ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) |
16 | 15 | rgen2w 3148 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧)) |
17 | df-po 5467 | . 2 ⊢ ( [⊊] Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥 [⊊] 𝑥 ∧ ((𝑥 [⊊] 𝑦 ∧ 𝑦 [⊊] 𝑧) → 𝑥 [⊊] 𝑧))) | |
18 | 16, 17 | mpbir 232 | 1 ⊢ [⊊] Po 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wral 3135 ⊊ wpss 3934 class class class wbr 5057 Po wpo 5465 [⊊] crpss 7437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-po 5467 df-xp 5554 df-rel 5555 df-rpss 7438 |
This theorem is referenced by: sorpss 7443 fin23lem40 9761 isfin1-3 9796 zorng 9914 fin2so 34760 |
Copyright terms: Public domain | W3C validator |