Theorem suplocexprlemss 7890

Description: Lemma for suplocexpr 7900. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))

Assertion

Ref	Expression
suplocexprlemss	⊢ (𝜑 → 𝐴 ⊆ P)

Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)

Proof of Theorem suplocexprlemss

Step	Hyp	Ref	Expression
1		suplocexpr.ub	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
2		rsp 2577	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦<P 𝑥))
3		ltrelpr 7680	. . . . . . . 8 ⊢ <P ⊆ (P × P)
4	3	brel 4768	. . . . . . 7 ⊢ (𝑦<P 𝑥 → (𝑦 ∈ P ∧ 𝑥 ∈ P))
5	4	simpld 112	. . . . . 6 ⊢ (𝑦<P 𝑥 → 𝑦 ∈ P)
6	2, 5	syl6 33	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦 ∈ P))
7	6	a1i 9	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦 ∈ P)))
8	7	rexlimdvw 2652	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦 ∈ P)))
9	1, 8	mpd 13	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝑦 ∈ P))
10	9	ssrdv 3230	1 ⊢ (𝜑 → 𝐴 ⊆ P)

Syntax hints:   → wi 4   ∨ wo 713  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3197   class class class wbr 4082  Pcnp 7466  <_P cltp 7470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4722  df-iltp 7645

This theorem is referenced by:  suplocexprlemml  7891  suplocexprlemrl  7892  suplocexprlemmu  7893  suplocexprlemru  7894  suplocexprlemdisj  7895  suplocexprlemloc  7896  suplocexprlemex  7897  suplocexprlemub  7898  suplocexprlemlub  7899