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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 6303. (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 4398 | . 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V)) | |
2 | on0eqel 6302 | . . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥)) | |
3 | 0nelxp 5583 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
4 | eleq1 2900 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V))) | |
5 | 3, 4 | mtbiri 329 | . . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V)) |
6 | 0nelelxp 5584 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥) | |
7 | 6 | con2i 141 | . . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V)) |
8 | 5, 7 | jaoi 853 | . . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V)) |
9 | 2, 8 | syl 17 | . 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V)) |
10 | 1, 9 | mprgbir 3153 | 1 ⊢ (On ∩ (V × V)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ∅c0 4290 × cxp 5547 Oncon0 6185 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-ord 6188 df-on 6189 |
This theorem is referenced by: (None) |
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