Theorem bj-peano4 14478

Description: Remove from peano4 4594 dependency on ax-setind 4534. 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 994	. . . . 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 14477	. . . . 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 4414	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ω → 𝐵 ∈ suc 𝐵)
6		eleq2 2241	. . . . . . . . . . . 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 4402	. . . . . . . . . . 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 1015	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
12		sucidg 4414	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
13		eleq2 2241	. . . . . . . . . . . 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 4402	. . . . . . . . . . 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 1016	. . . . . . . 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 2179	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	20	orbi2i 762	. . . . . . . 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 817	. . . . . 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 724	. . . 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 1205	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
29		suceq 4400	. 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 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  suc csuc 4363  ωcom 4587

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4127  ax-pr 4207  ax-un 4431  ax-bd0 14336  ax-bdor 14339  ax-bdn 14340  ax-bdal 14341  ax-bdex 14342  ax-bdeq 14343  ax-bdel 14344  ax-bdsb 14345  ax-bdsep 14407  ax-infvn 14464

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3809  df-int 3844  df-suc 4369  df-iom 4588  df-bdc 14364  df-bj-ind 14450

This theorem is referenced by: (None)