Theorem bj-peano4 16742

Description: Remove from peano4 4721 dependency on ax-setind 4661. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-peano4	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem bj-peano4

Step	Hyp	Ref	Expression
1		3simpa 1021	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
2		pm3.22 265	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ ω ∧ 𝐴 ∈ ω))
3		bj-nnen2lp 16741	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ¬ (𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵))
4	1, 2, 3	3syl 17	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ¬ (𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵))
5		sucidg 4539	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ω → 𝐵 ∈ suc 𝐵)
6		eleq2 2298	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
7	5, 6	syl5ibrcom 157	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
8		elsucg 4527	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
9	7, 8	sylibd 149	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
10	9	imp 124	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
11	10	3adant1 1042	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
12		sucidg 4539	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
13		eleq2 2298	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
14	12, 13	syl5ibcom 155	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
15		elsucg 4527	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
16	14, 15	sylibd 149	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
17	16	imp 124	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
18	17	3adant2 1043	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
19	11, 18	jca 306	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
20		eqcom 2236	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	20	orbi2i 770	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ↔ (𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
22	21	anbi1i 458	. . . . . . 7 ⊢ (((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) ↔ ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
23	19, 22	sylib 122	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
24		ordir 825	. . . . . 6 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
25	23, 24	sylibr 134	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) ∨ 𝐴 = 𝐵))
26	25	ord 732	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 = 𝐵))
27	4, 26	mpd 13	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
28	27	3expia 1232	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
29		suceq 4525	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
30	28, 29	impbid1 142	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  suc csuc 4488  ωcom 4714

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-nul 4238  ax-pr 4324  ax-un 4556  ax-bd0 16600  ax-bdor 16603  ax-bdn 16604  ax-bdal 16605  ax-bdex 16606  ax-bdeq 16607  ax-bdel 16608  ax-bdsb 16609  ax-bdsep 16671  ax-infvn 16728

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-suc 4494  df-iom 4715  df-bdc 16628  df-bj-ind 16714

This theorem is referenced by: (None)