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Mathbox for Rodolfo Medina |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem9 | Structured version Visualization version GIF version |
Description: Lemma for prter3 38481. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
Ref | Expression |
---|---|
prtlem9 | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3220 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
2 | eceq1 8763 | . . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
3 | 2 | reximi 3073 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
4 | 1, 3 | sylbi 216 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 [cec 8723 |
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-ext 2696 |
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-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 |
This theorem is referenced by: (None) |
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