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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 9111 | . . 3 ⊢ ((𝐶 ≺ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝐶 ≺ 𝐵) | |
2 | 1 | ancoms 460 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≺ 𝐴) → 𝐶 ≺ 𝐵) |
3 | ensym 8999 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | sdomentr 9111 | . . . 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 5149 ≈ cen 8936 ≺ csdm 8938 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 |
This theorem is referenced by: djuxpdom 10180 alephval2 10567 engch 10623 canthp1lem2 10648 hargch 10668 alephgch 10669 ovoliunnfl 36530 |
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