Theorem frind 30984

Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 30983). This principle states that if 𝐵 is a subclass of a founded class 𝐴 with the property that every element of 𝐵 whose initial segment is included in 𝐴 is itself equal to 𝐴. Compare wfi 5630 and tfi 6945, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
frind	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

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

Proof of Theorem frind

Step	Hyp	Ref	Expression
1		ssdif0 3896	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 2829	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		difss 3699	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
4		frmin 30983	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)
5		eldif 3550	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
6	5	anbi1i 727	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅))
7		anass 679	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)))
8		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ∧ ¬ 𝑦 ∈ 𝐵))
9		indif2 3829	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵)) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
10		df-pred 5597	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦}))
11		incom 3767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
12	10, 11	eqtri 2632	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
13		df-pred 5597	. . . . . . . . . . . . . . . . . . . 20 ⊢ Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (◡𝑅 “ {𝑦}))
14		incom 3767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
15	13, 14	eqtri 2632	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, 𝐴, 𝑦) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
16	15	difeq1i 3686	. . . . . . . . . . . . . . . . . 18 ⊢ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
17	9, 12, 16	3eqtr4i 2642	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
18	17	eqeq1i 2615	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
19		ssdif0 3896	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
20	18, 19	bitr4i 266	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
21	20	anbi1i 727	. . . . . . . . . . . . . 14 ⊢ ((Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ∧ ¬ 𝑦 ∈ 𝐵) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
22	8, 21	bitri 263	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
23	22	anbi2i 726	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
24	6, 7, 23	3bitri 285	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
25	24	rexbii2 3021	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
26		rexanali 2981	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
27	25, 26	bitri 263	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
28	4, 27	sylib 207	. . . . . . . 8 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
29	28	ex 449	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
30	3, 29	mpani 708	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ((𝐴 ∖ 𝐵) ≠ ∅ → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
31	2, 30	syl5bi 231	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (¬ 𝐴 ⊆ 𝐵 → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
32	31	con4d 113	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵) → 𝐴 ⊆ 𝐵))
33	32	imp 444	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)) → 𝐴 ⊆ 𝐵)
34	33	adantrl 748	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 ⊆ 𝐵)
35		simprl 790	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐵 ⊆ 𝐴)
36	34, 35	eqssd 3585	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   Fr wfr 4994   Se wse 4995  ^◡ccnv 5037   “ cima 5041  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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-trpred 30962

This theorem is referenced by:  frindi  30985  frinsg  30986