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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 8514 | . 2 ⊢ Rel ≈ | |
2 | inss1 4205 | . . 3 ⊢ ( ≼ ∩ ◡ ≼ ) ⊆ ≼ | |
3 | reldom 8515 | . . 3 ⊢ Rel ≼ | |
4 | relss 5656 | . . 3 ⊢ (( ≼ ∩ ◡ ≼ ) ⊆ ≼ → (Rel ≼ → Rel ( ≼ ∩ ◡ ≼ ))) | |
5 | 2, 3, 4 | mp2 9 | . 2 ⊢ Rel ( ≼ ∩ ◡ ≼ ) |
6 | brin 5118 | . . 3 ⊢ (𝑥( ≼ ∩ ◡ ≼ )𝑦 ↔ (𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦)) | |
7 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5753 | . . . 4 ⊢ (𝑥◡ ≼ 𝑦 ↔ 𝑦 ≼ 𝑥) |
10 | 9 | anbi2i 624 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑥◡ ≼ 𝑦) ↔ (𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥)) |
11 | sbthb 8638 | . . 3 ⊢ ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) ↔ 𝑥 ≈ 𝑦) | |
12 | 6, 10, 11 | 3bitrri 300 | . 2 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑥( ≼ ∩ ◡ ≼ )𝑦) |
13 | 1, 5, 12 | eqbrriv 5664 | 1 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 ◡ccnv 5554 Rel wrel 5560 ≈ cen 8506 ≼ cdom 8507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-er 8289 df-en 8510 df-dom 8511 |
This theorem is referenced by: dfsdom2 8640 |
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