Theorem frpoind 6345

Description: The principle of well-founded induction over a partial order. This theorem is a version of frind 9755 that does not require the axiom of infinity and can be used to prove wfi 6352 and tfi 7864. (Contributed by Scott Fenton, 11-Feb-2022.)

Assertion

Ref	Expression
frpoind	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 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 frpoind

Step	Hyp	Ref	Expression
1		ssdif0 4313	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 3002	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		difss 4072	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
4		frpomin2 6344	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)
5		eldif 4049	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
6	5	anbi1i 636	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅))
7		anass 474	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)))
8		indif2 4226	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵)) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
9		df-pred 6304	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦}))
10		incom 4106	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
11	9, 10	eqtri 2783	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
12		df-pred 6304	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (◡𝑅 “ {𝑦}))
13		incom 4106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
14	12, 13	eqtri 2783	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, 𝐴, 𝑦) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
15	14	difeq1i 4059	. . . . . . . . . . . . . . . . 17 ⊢ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
16	8, 11, 15	3eqtr4i 2793	. . . . . . . . . . . . . . . 16 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
17	16	eqeq1i 2765	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
18		ssdif0 4313	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
19	17, 18	bitr4i 281	. . . . . . . . . . . . . 14 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
20	19	anbi1ci 638	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
21	20	anbi2i 635	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
22	6, 7, 21	3bitri 300	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
23	22	rexbii2 3105	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
24		rexanali 3116	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
25	23, 24	bitri 278	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
26	4, 25	sylib 221	. . . . . . . 8 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
27	26	ex 418	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
28	3, 27	mpani 709	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ((𝐴 ∖ 𝐵) ≠ ∅ → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
29	2, 28	biimtrid 245	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (¬ 𝐴 ⊆ 𝐵 → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
30	29	con4d 116	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵) → 𝐴 ⊆ 𝐵))
31	30	imp 412	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)) → 𝐴 ⊆ 𝐵)
32	31	adantrl 729	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 ⊆ 𝐵)
33		simprl 783	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐵 ⊆ 𝐴)
34	32, 33	eqssd 3925	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  {csn 4583   Po wpo 5556   Fr wfr 5600   Se wse 5601  ^◡ccnv 5649   “ cima 5653  Predcpred 6303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-po 5558  df-fr 5603  df-se 5604  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304

This theorem is used by:  frpoinsg  6346  wfi  6352