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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem9 | Structured version Visualization version GIF version |
Description: Lemma for prter3 33185. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
Ref | Expression |
---|---|
prtlem9 | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3044 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
2 | eceq1 7669 | . . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
3 | 2 | reximi 2994 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
4 | 1, 3 | sylbi 206 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 [cec 7627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 |
This theorem is referenced by: (None) |
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