Theorem bj-peano4 10080

Description: Remove from peano4 4320 dependency on ax-setind 4262. 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 901	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
2		pm3.22 252	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ ω ∧ 𝐴 ∈ ω))
3		bj-nnen2lp 10079	. . . . 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 4153	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ω → 𝐵 ∈ suc 𝐵)
6		eleq2 2101	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐵))
7	5, 6	syl5ibrcom 146	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → (suc 𝐴 = suc 𝐵 → 𝐵 ∈ suc 𝐴))
8		elsucg 4141	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
9	7, 8	sylibd 138	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (suc 𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)))
10	9	imp 115	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
11	10	3adant1 922	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
12		sucidg 4153	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
13		eleq2 2101	. . . . . . . . . . . 12 ⊢ (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ suc 𝐵))
14	12, 13	syl5ibcom 144	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (suc 𝐴 = suc 𝐵 → 𝐴 ∈ suc 𝐵))
15		elsucg 4141	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
16	14, 15	sylibd 138	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (suc 𝐴 = suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
17	16	imp 115	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
18	17	3adant2 923	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
19	11, 18	jca 290	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
20		eqcom 2042	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
21	20	orbi2i 679	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ↔ (𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
22	21	anbi1i 431	. . . . . . 7 ⊢ (((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) ↔ ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
23	19, 22	sylib 127	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
24		ordir 730	. . . . . 6 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵) ∧ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
25	23, 24	sylibr 137	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) ∨ 𝐴 = 𝐵))
26	25	ord 643	. . . 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 1106	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 → 𝐴 = 𝐵))
29		suceq 4139	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
30	28, 29	impbid1 130	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  suc csuc 4102  ωcom 4313

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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051

This theorem is referenced by: (None)