Theorem frirr 5594

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 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝑅 Fr 𝐴)
2		snssi 4717	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
3	2	adantl 482	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
4		snnzg 4706	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ≠ ∅)
5	4	adantl 482	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ≠ ∅)
6		snex 5368	. . . 4 ⊢ {𝐵} ∈ V
7	6	frc 5581	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
8	1, 3, 5, 7	syl3anc 1379	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9		breq1 5075	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
10	9	rabeq0w 4315	. . . . . 6 ⊢ ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦)
11		breq2 5076	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐵))
12	11	notbid 319	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐵))
13	12	ralbidv 3162	. . . . . 6 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
14	10, 13	bitrid 284	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
15	14	rexsng 4608	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
16		breq1 5075	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝐵 ↔ 𝐵𝑅𝐵))
17	16	notbid 319	. . . . 5 ⊢ (𝑧 = 𝐵 → (¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
18	17	ralsng 4607	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
19	15, 18	bitrd 280	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
20	19	adantl 482	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
21	8, 20	mpbid 233	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  {crab 3391   ⊆ wss 3883  ∅c0 4261  {csn 4555   class class class wbr 5072   Fr wfr 5568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-fr 5571

This theorem is referenced by:  efrirr  5598  predfrirr  6285  dfwe2  7717  bnj1417  35223  efrunt  35941  ifr0  44893