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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem11 | Structured version Visualization version GIF version |
Description: Lemma for prter2 35499. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Ref | Expression |
---|---|
prtlem11 | ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq1 8125 | . . . 4 ⊢ (𝑥 = 𝐶 → [𝑥] ∼ = [𝐶] ∼ ) | |
2 | 1 | rspceeqv 3546 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ) |
3 | elqsg 8146 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ (𝐴 / ∼ ) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )) | |
4 | 2, 3 | syl5ibr 238 | . 2 ⊢ (𝐵 ∈ 𝐷 → ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 ∈ (𝐴 / ∼ ))) |
5 | 4 | expd 408 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∃wrex 3082 [cec 8085 / cqs 8086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-xp 5409 df-cnv 5411 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ec 8089 df-qs 8093 |
This theorem is referenced by: prter2 35499 |
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