Theorem bj-nnen2lp 15600

Description: A version of en2lp 4590 for natural numbers, which does not require ax-setind 4573.

Note: using this theorem and bj-nnelirr 15599, one can remove dependency on ax-setind 4573 from nntri2 6552 and nndcel 6558; 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 15599	. . 3 ⊢ (𝐵 ∈ ω → ¬ 𝐵 ∈ 𝐵)
2	1	adantl 277	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ 𝐵 ∈ 𝐵)
3		bj-nntrans 15597	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
4	3	adantl 277	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
5		ssel 3177	. . . 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 664	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2167   ⊆ wss 3157  ωcom 4626

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4159  ax-pr 4242  ax-un 4468  ax-bd0 15459  ax-bdor 15462  ax-bdn 15463  ax-bdal 15464  ax-bdex 15465  ax-bdeq 15466  ax-bdel 15467  ax-bdsb 15468  ax-bdsep 15530  ax-infvn 15587

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627  df-bdc 15487  df-bj-ind 15573

This theorem is referenced by:  bj-peano4  15601