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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 5114 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
2 | opelxp1 5074 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (V × V) → 𝐴 ∈ V) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝐴 ∈ V) |
4 | 3 | anim1i 590 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
5 | 4 | ancoms 468 | . 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
6 | opelopab3.3 | . . . . 5 ⊢ (𝜒 → 𝐴 ∈ 𝐶) | |
7 | elex 3185 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜒 → 𝐴 ∈ V) |
9 | 8 | anim1i 590 | . . 3 ⊢ ((𝜒 ∧ 𝐵 ∈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
10 | 9 | ancoms 468 | . 2 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝜒) → (𝐴 ∈ V ∧ 𝐵 ∈ 𝐷)) |
11 | opelopab3.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
12 | opelopab3.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
13 | 11, 12 | opelopabg 4918 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
14 | 5, 10, 13 | pm5.21nd 939 | 1 ⊢ (𝐵 ∈ 𝐷 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 {copab 4642 × cxp 5036 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-eu 2462 df-mo 2463 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-opab 4644 df-xp 5044 |
This theorem is referenced by: (None) |
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