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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 8562 | . . 3 ⊢ ((𝐶 ≼ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝐶 ≼ 𝐵) | |
2 | 1 | ancoms 461 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼ 𝐴) → 𝐶 ≼ 𝐵) |
3 | ensym 8552 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | domentr 8562 | . . . 4 ⊢ ((𝐶 ≼ 𝐵 ∧ 𝐵 ≈ 𝐴) → 𝐶 ≼ 𝐴) | |
5 | 4 | ancoms 461 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐶 ≼ 𝐵) → 𝐶 ≼ 𝐴) |
6 | 3, 5 | sylan 582 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼ 𝐵) → 𝐶 ≼ 𝐴) |
7 | 2, 6 | impbida 799 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 class class class wbr 5059 ≈ cen 8500 ≼ cdom 8501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-er 8283 df-en 8504 df-dom 8505 |
This theorem is referenced by: infdiffi 9115 carddomi2 9393 numdom 9458 djudom2 9603 infdif 9625 fin45 9808 fin67 9811 aleph1 9987 gchdomtri 10045 gchpwdom 10086 gchhar 10095 ctbnfien 39408 |
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