Theorem wsuclb 34788

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 5877	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌))
5	2, 3, 4	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌))
6	1, 5	mpbird 256	. . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋)
7		elpredg 6311	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋))
8	3, 2, 7	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋))
9	6, 8	mpbird 256	. . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋))
10		wsuclb.1	. . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴)
11		weso 5666	. . . . 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 5151	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
15	14	rspcev 3612	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
16	2, 1, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
17	10, 13, 3, 16	wsuclem 34785	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
18	12, 17	inflb 9480	. . 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 34772	. . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
21	20	breq2i 5155	. 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
22	19, 21	sylnibr 328	1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∈ wcel 2106  ∃wrex 3070   class class class wbr 5147   Or wor 5586   Se wse 5628   We wwe 5629  ^◡ccnv 5674  Predcpred 6296  infcinf 9432  wsuccwsuc 34770

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-iota 6492  df-riota 7361  df-sup 9433  df-inf 9434  df-wsuc 34772

This theorem is referenced by: (None)