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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 6511. (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 4456 | . 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V)) | |
2 | on0eqel 6510 | . . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥)) | |
3 | 0nelxp 5723 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
4 | eleq1 2827 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V))) | |
5 | 3, 4 | mtbiri 327 | . . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V)) |
6 | 0nelelxp 5724 | . . . . 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 3066 | 1 ⊢ (On ∩ (V × V)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ∅c0 4339 × cxp 5687 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-ord 6389 df-on 6390 |
This theorem is referenced by: (None) |
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