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| Mirrors > Home > MPE Home > Th. List > opwo0id | Structured version Visualization version GIF version | ||
| Description: An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.) | 
| Ref | Expression | 
|---|---|
| opwo0id | ⊢ 〈𝑋, 𝑌〉 = (〈𝑋, 𝑌〉 ∖ {∅}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0nelop 5501 | . . . 4 ⊢ ¬ ∅ ∈ 〈𝑋, 𝑌〉 | |
| 2 | disjsn 4711 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 〈𝑋, 𝑌〉) | |
| 3 | 1, 2 | mpbir 231 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∩ {∅}) = ∅ | 
| 4 | disjdif2 4480 | . . 3 ⊢ ((〈𝑋, 𝑌〉 ∩ {∅}) = ∅ → (〈𝑋, 𝑌〉 ∖ {∅}) = 〈𝑋, 𝑌〉) | |
| 5 | 3, 4 | ax-mp 5 | . 2 ⊢ (〈𝑋, 𝑌〉 ∖ {∅}) = 〈𝑋, 𝑌〉 | 
| 6 | 5 | eqcomi 2746 | 1 ⊢ 〈𝑋, 𝑌〉 = (〈𝑋, 𝑌〉 ∖ {∅}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∩ cin 3950 ∅c0 4333 {csn 4626 〈cop 4632 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 | 
| This theorem is referenced by: fundmge2nop0 14541 | 
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