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Theorem bnj563 30067
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 30031 . . . . 5 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3 bnj252 30022 . . . . 5 ((𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
42, 3bitri 263 . . . 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 206 . 2 (𝜂𝑛 = suc 𝑚)
7 bnj563.21 . . . 4 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
87simp2bi 1070 . . 3 (𝜌 → suc 𝑖𝑛)
97simp3bi 1071 . . 3 (𝜌𝑚 ≠ suc 𝑖)
108, 9jca 553 . 2 (𝜌 → (suc 𝑖𝑛𝑚 ≠ suc 𝑖))
11 necom 2835 . . . 4 (𝑚 ≠ suc 𝑖 ↔ suc 𝑖𝑚)
12 eleq2 2677 . . . . . 6 (𝑛 = suc 𝑚 → (suc 𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑚))
1312biimpa 500 . . . . 5 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛) → suc 𝑖 ∈ suc 𝑚)
14 elsuci 5708 . . . . . . 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 2873 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1715, 16bitr3i 265 . . . . . . 7 ((suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1814, 17sylib 207 . . . . . 6 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 → suc 𝑖𝑚))
1918imp 444 . . . . 5 ((suc 𝑖 ∈ suc 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2013, 19stoic3 1692 . . . 4 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2111, 20syl3an3b 1356 . . 3 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) → suc 𝑖𝑚)
22213expb 1258 . 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 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  suc csuc 5642  ωcom 6957  w-bnj17 30005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-sn 4126  df-suc 5646  df-bnj17 30006
This theorem is referenced by:  bnj570  30229
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