Theorem bnj563 32016

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj563.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj563.21	⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))

Assertion

Ref	Expression
bnj563	⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)

Proof of Theorem bnj563

Step	Hyp	Ref	Expression
1		bnj563.19	. . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2		bnj312 31984	. . . . 5 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ 𝑚 ∈ 𝐷 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3		bnj252 31975	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 ∈ 𝐷 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚 ∈ 𝐷 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
4	2, 3	bitri 277	. . . 4 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚 ∈ 𝐷 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
5	4	simplbi 500	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑛 = suc 𝑚)
6	1, 5	sylbi 219	. 2 ⊢ (𝜂 → 𝑛 = suc 𝑚)
7		bnj563.21	. . . 4 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
8	7	simp2bi 1142	. . 3 ⊢ (𝜌 → suc 𝑖 ∈ 𝑛)
9	7	simp3bi 1143	. . 3 ⊢ (𝜌 → 𝑚 ≠ suc 𝑖)
10	8, 9	jca 514	. 2 ⊢ (𝜌 → (suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
11		necom 3071	. . . 4 ⊢ (𝑚 ≠ suc 𝑖 ↔ suc 𝑖 ≠ 𝑚)
12		eleq2 2903	. . . . . 6 ⊢ (𝑛 = suc 𝑚 → (suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ suc 𝑚))
13	12	biimpa 479	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ suc 𝑖 ∈ 𝑛) → suc 𝑖 ∈ suc 𝑚)
14		elsuci 6259	. . . . . . 7 ⊢ (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖 ∈ 𝑚 ∨ suc 𝑖 = 𝑚))
15		orcom 866	. . . . . . . 8 ⊢ ((suc 𝑖 = 𝑚 ∨ suc 𝑖 ∈ 𝑚) ↔ (suc 𝑖 ∈ 𝑚 ∨ suc 𝑖 = 𝑚))
16		neor 3110	. . . . . . . 8 ⊢ ((suc 𝑖 = 𝑚 ∨ suc 𝑖 ∈ 𝑚) ↔ (suc 𝑖 ≠ 𝑚 → suc 𝑖 ∈ 𝑚))
17	15, 16	bitr3i 279	. . . . . . 7 ⊢ ((suc 𝑖 ∈ 𝑚 ∨ suc 𝑖 = 𝑚) ↔ (suc 𝑖 ≠ 𝑚 → suc 𝑖 ∈ 𝑚))
18	14, 17	sylib 220	. . . . . 6 ⊢ (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖 ≠ 𝑚 → suc 𝑖 ∈ 𝑚))
19	18	imp 409	. . . . 5 ⊢ ((suc 𝑖 ∈ suc 𝑚 ∧ suc 𝑖 ≠ 𝑚) → suc 𝑖 ∈ 𝑚)
20	13, 19	stoic3 1777	. . . 4 ⊢ ((𝑛 = suc 𝑚 ∧ suc 𝑖 ∈ 𝑛 ∧ suc 𝑖 ≠ 𝑚) → suc 𝑖 ∈ 𝑚)
21	11, 20	syl3an3b 1401	. . 3 ⊢ ((𝑛 = suc 𝑚 ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖) → suc 𝑖 ∈ 𝑚)
22	21	3expb 1116	. 2 ⊢ ((𝑛 = suc 𝑚 ∧ (suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖)) → suc 𝑖 ∈ 𝑚)
23	6, 10, 22	syl2an 597	1 ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  suc csuc 6195  ωcom 7582   ∧ w-bnj17 31958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-un 3943  df-sn 4570  df-suc 6199  df-bnj17 31959

This theorem is referenced by:  bnj570  32179