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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 35549. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
Ref | Expression |
---|---|
prtlem9 | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3231 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴) | |
2 | eceq1 8177 | . . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
3 | 2 | reximi 3207 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
4 | 1, 3 | sylbi 218 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 [cec 8137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-ec 8141 |
This theorem is referenced by: (None) |
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