Theorem tz6.26 5628

Description: All nonempty (possibly proper) subclasses of 𝐴, which has a well-founded relation 𝑅, have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
tz6.26	⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem tz6.26

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wereu2 5035	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
2		reurex 3137	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦 → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
3	1, 2	syl 17	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
4		rabeq0 3911	. . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
5		dfrab3 3861	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = (𝐵 ∩ {𝑥 ∣ 𝑥𝑅𝑦})
6		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
7	6	dfpred2 5606	. . . . . 6 ⊢ Pred(𝑅, 𝐵, 𝑦) = (𝐵 ∩ {𝑥 ∣ 𝑥𝑅𝑦})
8	5, 7	eqtr4i 2635	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = Pred(𝑅, 𝐵, 𝑦)
9	8	eqeq1i 2615	. . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = ∅ ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
10	4, 9	bitr3i 265	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ Pred(𝑅, 𝐵, 𝑦) = ∅)
11	10	rexbii 3023	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
12	3, 11	sylib 207	1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   Se wse 4995   We wwe 4996  Predcpred 5596

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597

This theorem is referenced by:  tz6.26i  5629  wfi  5630  wzel  31015  wzelOLD  31016  wsuclem  31017  wsuclemOLD  31018