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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 6302. (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 4395 | . 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V)) | |
2 | on0eqel 6301 | . . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥)) | |
3 | 0nelxp 5582 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
4 | eleq1 2897 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V))) | |
5 | 3, 4 | mtbiri 328 | . . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V)) |
6 | 0nelelxp 5583 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥) | |
7 | 6 | con2i 141 | . . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V)) |
8 | 5, 7 | jaoi 851 | . . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V)) |
9 | 2, 8 | syl 17 | . 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V)) |
10 | 1, 9 | mprgbir 3150 | 1 ⊢ (On ∩ (V × V)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ∅c0 4288 × cxp 5546 Oncon0 6184 |
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 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-ord 6187 df-on 6188 |
This theorem is referenced by: (None) |
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