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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 5377 | . . . 4 ⊢ ¬ ∅ ∈ 〈𝑋, 𝑌〉 | |
2 | disjsn 4639 | . . . 4 ⊢ ((〈𝑋, 𝑌〉 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 〈𝑋, 𝑌〉) | |
3 | 1, 2 | mpbir 232 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∩ {∅}) = ∅ |
4 | disjdif2 4424 | . . 3 ⊢ ((〈𝑋, 𝑌〉 ∩ {∅}) = ∅ → (〈𝑋, 𝑌〉 ∖ {∅}) = 〈𝑋, 𝑌〉) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (〈𝑋, 𝑌〉 ∖ {∅}) = 〈𝑋, 𝑌〉 |
6 | 5 | eqcomi 2827 | 1 ⊢ 〈𝑋, 𝑌〉 = (〈𝑋, 𝑌〉 ∖ {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∩ cin 3932 ∅c0 4288 {csn 4557 〈cop 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 |
This theorem is referenced by: fundmge2nop0 13838 |
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