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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nnen2lp | GIF version |
Description: A version of en2lp 4232 for natural numbers, which does not require
ax-setind 4220.
Note: using this theorem and bj-nnelirr 9413, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnen2lp | ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ (A ∈ B ∧ B ∈ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnelirr 9413 | . . 3 ⊢ (B ∈ 𝜔 → ¬ B ∈ B) | |
2 | 1 | adantl 262 | . 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ B ∈ B) |
3 | bj-nntrans 9411 | . . . . 5 ⊢ (B ∈ 𝜔 → (A ∈ B → A ⊆ B)) | |
4 | 3 | adantl 262 | . . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B → A ⊆ B)) |
5 | ssel 2933 | . . . 4 ⊢ (A ⊆ B → (B ∈ A → B ∈ B)) | |
6 | 4, 5 | syl6 29 | . . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B → (B ∈ A → B ∈ B))) |
7 | 6 | impd 242 | . 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A ∈ B ∧ B ∈ A) → B ∈ B)) |
8 | 2, 7 | mtod 588 | 1 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ (A ∈ B ∧ B ∈ A)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∈ wcel 1390 ⊆ wss 2911 𝜔com 4256 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdor 9271 ax-bdn 9272 ax-bdal 9273 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 ax-infvn 9401 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: bj-peano4 9415 |
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