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| Mirrors > Home > MPE Home > Th. List > onxpdisj | Structured version Visualization version GIF version | ||
| Description: Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 6508. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) | 
| Ref | Expression | 
|---|---|
| onxpdisj | ⊢ (On ∩ (V × V)) = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | disj 4449 | . 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V)) | |
| 2 | on0eqel 6507 | . . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥)) | |
| 3 | 0nelxp 5718 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
| 4 | eleq1 2828 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V))) | |
| 5 | 3, 4 | mtbiri 327 | . . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V)) | 
| 6 | 0nelelxp 5719 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥) | |
| 7 | 6 | con2i 139 | . . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V)) | 
| 8 | 5, 7 | jaoi 857 | . . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V)) | 
| 9 | 2, 8 | syl 17 | . 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V)) | 
| 10 | 1, 9 | mprgbir 3067 | 1 ⊢ (On ∩ (V × V)) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ∅c0 4332 × cxp 5682 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: (None) | 
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