Theorem bj-nnen2lp 16247

Description: A version of en2lp 4643 for natural numbers, which does not require ax-setind 4626.

Note: using this theorem and bj-nnelirr 16246, one can remove dependency on ax-setind 4626 from nntri2 6630 and nndcel 6636; 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 16246	. . 3 ⊢ (𝐵 ∈ ω → ¬ 𝐵 ∈ 𝐵)
2	1	adantl 277	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ 𝐵 ∈ 𝐵)
3		bj-nntrans 16244	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
4	3	adantl 277	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
5		ssel 3218	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
6	4, 5	syl6 33	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵)))
7	6	impd 254	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐵))
8	2, 7	mtod 667	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2200   ⊆ wss 3197  ωcom 4679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4209  ax-pr 4292  ax-un 4521  ax-bd0 16106  ax-bdor 16109  ax-bdn 16110  ax-bdal 16111  ax-bdex 16112  ax-bdeq 16113  ax-bdel 16114  ax-bdsb 16115  ax-bdsep 16177  ax-infvn 16234

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-suc 4459  df-iom 4680  df-bdc 16134  df-bj-ind 16220

This theorem is referenced by:  bj-peano4  16248