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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 6332. (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 4362 | . 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V)) | |
2 | on0eqel 6331 | . . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥)) | |
3 | 0nelxp 5585 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
4 | eleq1 2825 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V))) | |
5 | 3, 4 | mtbiri 330 | . . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V)) |
6 | 0nelelxp 5586 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥) | |
7 | 6 | con2i 141 | . . . 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 3076 | 1 ⊢ (On ∩ (V × V)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∩ cin 3865 ∅c0 4237 × cxp 5549 Oncon0 6213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-ord 6216 df-on 6217 |
This theorem is referenced by: (None) |
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