Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbthcl | Structured version Visualization version GIF version |
Description: Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.) |
Ref | Expression |
---|---|
sbthcl | ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8609 | . 2 ⊢ Rel ≈ | |
2 | inss1 4129 | . . 3 ⊢ ( ≼ ∩ ◡ ≼ ) ⊆ ≼ | |
3 | reldom 8610 | . . 3 ⊢ Rel ≼ | |
4 | relss 5638 | . . 3 ⊢ (( ≼ ∩ ◡ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ◡ ≼ ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≼ ∩ ◡ ≼ ) |
6 | brin 5091 | . . 3 ⊢ (𝑥( ≼ ∩ ◡ ≼ )𝑦 ↔ (𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦)) | |
7 | vex 3402 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | vex 3402 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5736 | . . . 4 ⊢ (𝑥◡ ≼ 𝑦 ↔ 𝑦 ≼ 𝑥) |
10 | 9 | anbi2i 626 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦) ↔ (𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥)) |
11 | sbthb 8745 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) ↔ 𝑥 ≈ 𝑦) | |
12 | 6, 10, 11 | 3bitrri 301 | . 2 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑥( ≼ ∩ ◡ ≼ )𝑦) |
13 | 1, 5, 12 | eqbrriv 5646 | 1 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∩ cin 3852 ⊆ wss 3853 class class class wbr 5039 ◡ccnv 5535 Rel wrel 5541 ≈ cen 8601 ≼ cdom 8602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-er 8369 df-en 8605 df-dom 8606 |
This theorem is referenced by: dfsdom2 8747 |
Copyright terms: Public domain | W3C validator |