Theorem wfi 6315

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 5622	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3		weso 5623	. . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4		sopo 5559	. . . . 5 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴)
6	5	adantr 480	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7		simpr 484	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8	2, 6, 7	3jca 1129	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴))
9		frpoind 6308	. 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 581	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903   Po wpo 5538   Or wor 5539   Fr wfr 5582   Se wse 5583   We wwe 5584  Predcpred 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267

This theorem is referenced by:  wfii  6316