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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 5236 | . . . 4 ⊢ ¬ ∅ ∈ 〈𝑋, 𝑌〉 | |
2 | disjsn 4515 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 〈𝑋, 𝑌〉) | |
3 | 1, 2 | mpbir 223 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∩ {∅}) = ∅ |
4 | disjdif2 4305 | . . 3 ⊢ ((〈𝑋, 𝑌〉 ∩ {∅}) = ∅ → (〈𝑋, 𝑌〉 ∖ {∅}) = 〈𝑋, 𝑌〉) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (〈𝑋, 𝑌〉 ∖ {∅}) = 〈𝑋, 𝑌〉 |
6 | 5 | eqcomi 2781 | 1 ⊢ 〈𝑋, 𝑌〉 = (〈𝑋, 𝑌〉 ∖ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1507 ∈ wcel 2050 ∖ cdif 3820 ∩ cin 3822 ∅c0 4172 {csn 4435 〈cop 4441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 |
This theorem is referenced by: fundmge2nop0 13655 |
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