Theorem bj-nnen2lp 9414

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.)

Assertion

Ref	Expression
bj-nnen2lp	⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ (A ∈ B ∧ B ∈ A))

Proof of Theorem bj-nnen2lp

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))

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