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Theorem bnj563 29873
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 29837 . . . . 5 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3 bnj252 29828 . . . . 5 ((𝑛 = suc 𝑚𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
42, 3bitri 262 . . . 4 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑛 = suc 𝑚 ∧ (𝑚𝐷𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
54simplbi 474 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑛 = suc 𝑚)
61, 5sylbi 205 . 2 (𝜂𝑛 = suc 𝑚)
7 bnj563.21 . . . 4 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
87simp2bi 1069 . . 3 (𝜌 → suc 𝑖𝑛)
97simp3bi 1070 . . 3 (𝜌𝑚 ≠ suc 𝑖)
108, 9jca 552 . 2 (𝜌 → (suc 𝑖𝑛𝑚 ≠ suc 𝑖))
11 necom 2834 . . . 4 (𝑚 ≠ suc 𝑖 ↔ suc 𝑖𝑚)
12 eleq2 2676 . . . . . 6 (𝑛 = suc 𝑚 → (suc 𝑖𝑛 ↔ suc 𝑖 ∈ suc 𝑚))
1312biimpa 499 . . . . 5 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛) → suc 𝑖 ∈ suc 𝑚)
14 elsuci 5694 . . . . . . 7 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚))
15 orcom 400 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚))
16 neor 2872 . . . . . . . 8 ((suc 𝑖 = 𝑚 ∨ suc 𝑖𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1715, 16bitr3i 264 . . . . . . 7 ((suc 𝑖𝑚 ∨ suc 𝑖 = 𝑚) ↔ (suc 𝑖𝑚 → suc 𝑖𝑚))
1814, 17sylib 206 . . . . . 6 (suc 𝑖 ∈ suc 𝑚 → (suc 𝑖𝑚 → suc 𝑖𝑚))
1918imp 443 . . . . 5 ((suc 𝑖 ∈ suc 𝑚 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2013, 19stoic3 1691 . . . 4 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛 ∧ suc 𝑖𝑚) → suc 𝑖𝑚)
2111, 20syl3an3b 1355 . . 3 ((𝑛 = suc 𝑚 ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) → suc 𝑖𝑚)
22213expb 1257 . 2 ((𝑛 = suc 𝑚 ∧ (suc 𝑖𝑛𝑚 ≠ suc 𝑖)) → suc 𝑖𝑚)
236, 10, 22syl2an 492 1 ((𝜂𝜌) → suc 𝑖𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  suc csuc 5628  ωcom 6934  w-bnj17 29811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-v 3174  df-un 3544  df-sn 4125  df-suc 5632  df-bnj17 29812
This theorem is referenced by:  bnj570  30035
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