Theorem wfi 6252

Description: The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)

Assertion

Ref	Expression
wfi	⊢ (((𝑅 We 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 wfi

Step	Hyp	Ref	Expression
1		wefr 5580	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2	1	adantr 481	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3		weso 5581	. . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4		sopo 5523	. . . . 5 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴)
6	5	adantr 481	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7		simpr 485	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8	2, 6, 7	3jca 1127	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴))
9		frpoind 6244	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)
10	8, 9	sylan 580	1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ∀wral 3066   ⊆ wss 3892   Po wpo 5502   Or wor 5503   Fr wfr 5542   Se wse 5543   We wwe 5544  Predcpred 6200

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201

This theorem is referenced by:  wfii  6254  wfisgOLD  6256