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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 7846 | . 2 ⊢ Rel ≈ | |
2 | inss1 3795 | . . 3 ⊢ ( ≼ ∩ ◡ ≼ ) ⊆ ≼ | |
3 | reldom 7847 | . . 3 ⊢ Rel ≼ | |
4 | relss 5129 | . . 3 ⊢ (( ≼ ∩ ◡ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ◡ ≼ ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≼ ∩ ◡ ≼ ) |
6 | brin 4634 | . . 3 ⊢ (𝑥( ≼ ∩ ◡ ≼ )𝑦 ↔ (𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦)) | |
7 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5227 | . . . 4 ⊢ (𝑥◡ ≼ 𝑦 ↔ 𝑦 ≼ 𝑥) |
10 | 9 | anbi2i 726 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦) ↔ (𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥)) |
11 | sbthb 7966 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) ↔ 𝑥 ≈ 𝑦) | |
12 | 6, 10, 11 | 3bitrri 286 | . 2 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑥( ≼ ∩ ◡ ≼ )𝑦) |
13 | 1, 5, 12 | eqbrriv 5138 | 1 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 ◡ccnv 5037 Rel wrel 5043 ≈ cen 7838 ≼ cdom 7839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 |
This theorem is referenced by: dfsdom2 7968 |
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