Theorem frirr 5380

Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
frirr	⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem frirr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 475	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝑅 Fr 𝐴)
2		snssi 4611	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
3	2	adantl 474	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
4		snnzg 4580	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ≠ ∅)
5	4	adantl 474	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ≠ ∅)
6		snex 5184	. . . 4 ⊢ {𝐵} ∈ V
7	6	frc 5369	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
8	1, 3, 5, 7	syl3anc 1352	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9		rabeq0 4218	. . . . . 6 ⊢ ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦)
10		breq2 4929	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
11	10	notbid 310	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
12	11	ralbidv 3140	. . . . . 6 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
13	9, 12	syl5bb 275	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
14	13	rexsng 4484	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
15		breq1 4928	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐵 ↔ 𝐵𝑅𝐵))
16	15	notbid 310	. . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
17	16	ralsng 4483	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
18	14, 17	bitrd 271	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
19	18	adantl 474	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
20	8, 19	mpbid 224	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051   ≠ wne 2960  ∀wral 3081  ∃wrex 3082  {crab 3085   ⊆ wss 3822  ∅c0 4172  {csn 4435   class class class wbr 4925   Fr wfr 5359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-fr 5362

This theorem is referenced by:  efrirr  5384  predfrirr  6012  dfwe2  7310  bnj1417  31990  efrunt  32496  ifr0  40239