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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opelopab3 | Structured version Visualization version GIF version |
Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.) |
Ref | Expression |
---|---|
opelopab3.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopab3.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
opelopab3.3 | ⊢ (𝜒 → 𝐴 ∈ 𝐶) |
Ref | Expression |
---|---|
opelopab3 | ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopaelxp 5725 | . . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ∈ (V × V)) | |
2 | opelxp1 5678 | . . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) → 𝐴 ∈ V) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ V) |
4 | 3 | anim1i 616 | . . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
5 | 4 | ancoms 460 | . 2 ⊢ ((𝐵 ∈ 𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
6 | opelopab3.3 | . . . . 5 ⊢ (𝜒 → 𝐴 ∈ 𝐶) | |
7 | 6 | elexd 3467 | . . . 4 ⊢ (𝜒 → 𝐴 ∈ V) |
8 | 7 | anim1i 616 | . . 3 ⊢ ((𝜒 ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
9 | 8 | ancoms 460 | . 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝜒) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
10 | opelopab3.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
11 | opelopab3.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
12 | 10, 11 | opelopabg 5499 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)) |
13 | 5, 9, 12 | pm5.21nd 801 | 1 ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ⟨cop 4596 {copab 5171 × cxp 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-opab 5172 df-xp 5643 |
This theorem is referenced by: (None) |
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