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Theorem bnj563 30939
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 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj563.21 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
Assertion
Ref Expression
bnj563 ((𝜂𝜌) → suc 𝑖𝑚)

Proof of Theorem bnj563
StepHypRef Expression
1 bnj563.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2 bnj312 30906 . . . . 5 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3 bnj252 30897 . . . . 5 ((𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
42, 3bitri 264 . . . 4 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
54simplbi 475 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑛 = suc 𝑚)
61, 5sylbi 207 . 2 (𝜂𝑛 = suc 𝑚)
7 bnj563.21 . . . 4 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
87simp2bi 1097 . . 3 (𝜌 → suc 𝑖𝑛)
97simp3bi 1098 . . 3 (𝜌𝑚 ≠ suc 𝑖)
108, 9jca 553 . 2 (𝜌 → (suc 𝑖𝑛𝑚 ≠ suc 𝑖))
11 necom 2876 . . . 4 (𝑚 ≠ suc 𝑖 ↔ suc 𝑖𝑚)
12 eleq2 2719 . . . . . 6 (𝑛 = suc 𝑚 → (suc 𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑚))
1312biimpa 500 . . . . 5 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛) → suc 𝑖 ∈ suc 𝑚)
14 elsuci 5829 . . . . . . 7 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚))
15 orcom 401 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚))
16 neor 2914 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1715, 16bitr3i 266 . . . . . . 7 ((suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1814, 17sylib 208 . . . . . 6 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 → suc 𝑖𝑚))
1918imp 444 . . . . 5 ((suc 𝑖 ∈ suc 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2013, 19stoic3 1741 . . . 4 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2111, 20syl3an3b 1404 . . 3 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) → suc 𝑖𝑚)
22213expb 1285 . 2 ((𝑛 = suc 𝑚 ∧ (suc 𝑖𝑛𝑚 ≠ suc 𝑖)) → suc 𝑖𝑚)
236, 10, 22syl2an 493 1 ((𝜂𝜌) → suc 𝑖𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  suc csuc 5763  ωcom 7107  w-bnj17 30880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-un 3612  df-sn 4211  df-suc 5767  df-bnj17 30881
This theorem is referenced by:  bnj570  31101
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