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Theorem bnj563 32034
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 32002 . . . . 5 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3 bnj252 31993 . . . . 5 ((𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
42, 3bitri 278 . . . 4 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
54simplbi 501 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑛 = suc 𝑚)
61, 5sylbi 220 . 2 (𝜂𝑛 = suc 𝑚)
7 bnj563.21 . . . 4 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
87simp2bi 1143 . . 3 (𝜌 → suc 𝑖𝑛)
97simp3bi 1144 . . 3 (𝜌𝑚 ≠ suc 𝑖)
108, 9jca 515 . 2 (𝜌 → (suc 𝑖𝑛𝑚 ≠ suc 𝑖))
11 necom 3066 . . . 4 (𝑚 ≠ suc 𝑖 ↔ suc 𝑖𝑚)
12 eleq2 2904 . . . . . 6 (𝑛 = suc 𝑚 → (suc 𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑚))
1312biimpa 480 . . . . 5 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛) → suc 𝑖 ∈ suc 𝑚)
14 elsuci 6240 . . . . . . 7 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚))
15 orcom 867 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚))
16 neor 3104 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1715, 16bitr3i 280 . . . . . . 7 ((suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1814, 17sylib 221 . . . . . 6 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 → suc 𝑖𝑚))
1918imp 410 . . . . 5 ((suc 𝑖 ∈ suc 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2013, 19stoic3 1778 . . . 4 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2111, 20syl3an3b 1402 . . 3 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) → suc 𝑖𝑚)
22213expb 1117 . 2 ((𝑛 = suc 𝑚 ∧ (suc 𝑖𝑛𝑚 ≠ suc 𝑖)) → suc 𝑖𝑚)
236, 10, 22syl2an 598 1 ((𝜂𝜌) → suc 𝑖𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wne 3013  suc csuc 6176  ωcom 7565  w-bnj17 31976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3014  df-v 3481  df-un 3923  df-sn 4549  df-suc 6180  df-bnj17 31977
This theorem is referenced by:  bnj570  32197
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