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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 8941 | . 2 ⊢ Rel ≈ | |
2 | inss1 4228 | . . 3 ⊢ ( ≼ ∩ ◡ ≼ ) ⊆ ≼ | |
3 | reldom 8942 | . . 3 ⊢ Rel ≼ | |
4 | relss 5780 | . . 3 ⊢ (( ≼ ∩ ◡ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ◡ ≼ ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≼ ∩ ◡ ≼ ) |
6 | brin 5200 | . . 3 ⊢ (𝑥( ≼ ∩ ◡ ≼ )𝑦 ↔ (𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦)) | |
7 | vex 3479 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | vex 3479 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5881 | . . . 4 ⊢ (𝑥◡ ≼ 𝑦 ↔ 𝑦 ≼ 𝑥) |
10 | 9 | anbi2i 624 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦) ↔ (𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥)) |
11 | sbthb 9091 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) ↔ 𝑥 ≈ 𝑦) | |
12 | 6, 10, 11 | 3bitrri 298 | . 2 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑥( ≼ ∩ ◡ ≼ )𝑦) |
13 | 1, 5, 12 | eqbrriv 5790 | 1 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 ◡ccnv 5675 Rel wrel 5681 ≈ cen 8933 ≼ cdom 8934 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-er 8700 df-en 8937 df-dom 8938 |
This theorem is referenced by: dfsdom2 9093 |
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