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Mirrors > Home > MPE Home > Th. List > opeqex | Structured version Visualization version GIF version |
Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
Ref | Expression |
---|---|
opeqex | ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 3080 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (〈𝐴, 𝐵〉 ≠ ∅ ↔ 〈𝐶, 𝐷〉 ≠ ∅)) | |
2 | opnz 5367 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | opnz 5367 | . 2 ⊢ (〈𝐶, 𝐷〉 ≠ ∅ ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
4 | 1, 2, 3 | 3bitr3g 315 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: oteqex2 5391 oteqex 5392 epelgOLD 5469 |
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