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Mirrors > Home > MPE Home > Th. List > sdomen2 | Structured version Visualization version GIF version |
Description: Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.) |
Ref | Expression |
---|---|
sdomen2 | ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≺ 𝐴 ↔ 𝐶 ≺ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomentr 8650 | . . 3 ⊢ ((𝐶 ≺ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝐶 ≺ 𝐵) | |
2 | 1 | ancoms 461 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≺ 𝐴) → 𝐶 ≺ 𝐵) |
3 | ensym 8557 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | sdomentr 8650 | . . . 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 5065 ≈ cen 8505 ≺ csdm 8507 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 |
This theorem is referenced by: djuxpdom 9610 alephval2 9993 engch 10049 canthp1lem2 10074 hargch 10094 alephgch 10095 ovoliunnfl 34933 |
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