Theorem wsuclb 36008

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 5834	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝑉) → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌))
5	2, 3, 4	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑌◡𝑅𝑋 ↔ 𝑋𝑅𝑌))
6	1, 5	mpbird 257	. . . 4 ⊢ (𝜑 → 𝑌◡𝑅𝑋)
7		elpredg 6279	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋))
8	3, 2, 7	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ 𝑌◡𝑅𝑋))
9	6, 8	mpbird 257	. . 3 ⊢ (𝜑 → 𝑌 ∈ Pred(◡𝑅, 𝐴, 𝑋))
10		wsuclb.1	. . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴)
11		weso 5622	. . . . 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 5089	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
15	14	rspcev 3564	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋𝑅𝑌) → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
16	2, 1, 15	syl2anc 585	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦)
17	10, 13, 3, 16	wsuclem 36005	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 (∀𝑏 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑏𝑅𝑎 ∧ ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 → ∃𝑐 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑐𝑅𝑏)))
18	12, 17	inflb 9403	. . 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 35992	. . 3 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
21	20	breq2i 5093	. 2 ⊢ (𝑌𝑅wsuc(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅))
22	19, 21	sylnibr 329	1 ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2114  ∃wrex 3061   class class class wbr 5085   Or wor 5538   Se wse 5582   We wwe 5583  ^◡ccnv 5630  Predcpred 6264  infcinf 9354  wsuccwsuc 35990

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 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-riota 7324  df-sup 9355  df-inf 9356  df-wsuc 35992

This theorem is referenced by: (None)