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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1148 | Structured version Visualization version GIF version |
Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1148 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 3215 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 = 𝑋) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 𝑥 = 𝑋) |
3 | bnj93 30933 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
4 | eleq1 2689 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
5 | 4 | anbi2d 740 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))) |
6 | bnj602 30985 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅)) | |
7 | 6 | eleq1d 2686 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ( pred(𝑥, 𝐴, 𝑅) ∈ V ↔ pred(𝑋, 𝐴, 𝑅) ∈ V)) |
8 | 5, 7 | imbi12d 334 | . . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))) |
9 | 3, 8 | mpbii 223 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)) |
10 | 2, 9 | bnj593 30815 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑥((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)) |
11 | 10 | bnj937 30842 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)) |
12 | 11 | pm2.43i 52 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 predc-bnj14 30754 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-bnj14 30755 df-bnj13 30757 df-bnj15 30759 |
This theorem is referenced by: bnj1136 31065 bnj1413 31103 |
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