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Mirrors > Home > MPE Home > Th. List > domen2 | Structured version Visualization version GIF version |
Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Ref | Expression |
---|---|
domen2 | ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domentr 8960 | . . 3 ⊢ ((𝐶 ≼ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝐶 ≼ 𝐵) | |
2 | 1 | ancoms 460 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼ 𝐴) → 𝐶 ≼ 𝐵) |
3 | ensym 8950 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | domentr 8960 | . . . 4 ⊢ ((𝐶 ≼ 𝐵 ∧ 𝐵 ≈ 𝐴) → 𝐶 ≼ 𝐴) | |
5 | 4 | ancoms 460 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐶 ≼ 𝐵) → 𝐶 ≼ 𝐴) |
6 | 3, 5 | sylan 581 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼ 𝐵) → 𝐶 ≼ 𝐴) |
7 | 2, 6 | impbida 800 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 class class class wbr 5110 ≈ cen 8887 ≼ cdom 8888 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-er 8655 df-en 8891 df-dom 8892 |
This theorem is referenced by: infdiffi 9601 carddomi2 9913 numdom 9981 djudom2 10126 infdif 10152 fin45 10335 fin67 10338 aleph1 10514 gchdomtri 10572 gchpwdom 10613 gchhar 10622 ctbnfien 41170 |
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