Theorem bj-nnen2lp 7176

Description: A version of en2lp 4216 for natural numbers, which does not require ax-setind 4204.

Note: using this theorem and bj-nnelirr 7175, one can remove dependency on ax-setind 4204 from nntri2 5988 and nndcel 5991; 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 7175	. . 3 ⊢ (B ∈ 𝜔 → ¬ B ∈ B)
2	1	adantl 262	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ B ∈ B)
3		bj-nntrans 7173	. . . . 5 ⊢ (B ∈ 𝜔 → (A ∈ B → A ⊆ B))
4	3	adantl 262	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B → A ⊆ B))
5		ssel 2916	. . . 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 576	1 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ¬ (A ∈ B ∧ B ∈ A))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1374   ⊆ wss 2894  𝜔com 4240

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdn 7044  ax-bdal 7045  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111  ax-infvn 7163

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150

This theorem is referenced by:  bj-peano4  7177