Theorem wsuclb 33110

Description: A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)

Hypotheses

Ref	Expression
wsuclb.1	⊢ (𝜑 → 𝑅 We 𝐴)
wsuclb.2	⊢ (𝜑 → 𝑅 Se 𝐴)
wsuclb.3	⊢ (𝜑 → 𝑋 ∈ 𝑉)
wsuclb.4	⊢ (𝜑 → 𝑌 ∈ 𝐴)
wsuclb.5	⊢ (𝜑 → 𝑋𝑅𝑌)

Assertion

Ref	Expression
wsuclb	⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Proof of Theorem wsuclb

Dummy variables 𝑎 𝑏 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wsuclb.5	. . . . 5 ⊢ (𝜑 → 𝑋𝑅𝑌)
2		wsuclb.4	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
3		wsuclb.3	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		brcnvg 5745	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌))
5	2, 3, 4	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌))
6	1, 5	mpbird 259	. . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋)
7		elpredg 6157	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋))
8	3, 2, 7	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋))
9	6, 8	mpbird 259	. . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋))
10		wsuclb.1	. . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴)
11		weso 5541	. . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
13		wsuclb.2	. . . . 5 ⊢ (𝜑 → 𝑅 Se 𝐴)
14		breq2 5063	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
15	14	rspcev 3623	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
16	2, 1, 15	syl2anc 586	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
17	10, 13, 3, 16	wsuclem 33107	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
18	12, 17	inflb 8947	. . 3 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) → ¬ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
19	9, 18	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
20		df-wsuc 33094	. . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
21	20	breq2i 5067	. 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
22	19, 21	sylnibr 331	1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∈ wcel 2110  ∃wrex 3139   class class class wbr 5059   Or wor 5468   Se wse 5507   We wwe 5508  ^◡ccnv 5549  Predcpred 6142  infcinf 8899  wsuccwsuc 33092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-cnv 5558  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-iota 6309  df-riota 7108  df-sup 8900  df-inf 8901  df-wsuc 33094

This theorem is referenced by: (None)