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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 9142 | . . 3 ⊢ ((𝐶 ≺ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝐶 ≺ 𝐵) | |
2 | 1 | ancoms 457 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≺ 𝐴) → 𝐶 ≺ 𝐵) |
3 | ensym 9030 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | sdomentr 9142 | . . . 4 ⊢ ((𝐶 ≺ 𝐵 ∧ 𝐵 ≈ 𝐴) → 𝐶 ≺ 𝐴) | |
5 | 4 | ancoms 457 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐶 ≺ 𝐵) → 𝐶 ≺ 𝐴) |
6 | 3, 5 | sylan 578 | . 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 205 class class class wbr 5152 ≈ cen 8967 ≺ csdm 8969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 |
This theorem is referenced by: djuxpdom 10216 alephval2 10603 engch 10659 canthp1lem2 10684 hargch 10704 alephgch 10705 ovoliunnfl 37168 |
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