Theorem inecmo2 34121

Description: Lemma for ~? dfeldisj5 , and for ~? dfdisjs5 , ~? dfdisjALTV5 , ~? eldisjs5 (via inecmo3 34126, ~? cosscnvssid5 ). (Contributed by Peter Mazsa, 29-May-2018.) (Revised by Peter Mazsa, 2-Sep-2021.)

Assertion

Ref	Expression
inecmo2	⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))

Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑢,𝑅,𝑣,𝑥

Proof of Theorem inecmo2

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
2	1	inecmo 34120	. 2 ⊢ (Rel 𝑅 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥))
3	2	pm5.32ri 670	1 ⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483  ∀wral 2912  ∃*wrmo 2915   ∩ cin 3573  ∅c0 3915   class class class wbr 4653  Rel wrel 5119  [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  inecmo3  34126