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Theorem suplocexprlemss 7523

Description: Lemma for suplocexpr 7533. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)

**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 2480	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<_P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦<_P 𝑥))
3		ltrelpr 7313	. . . . . . . 8 ⊢ <_P ⊆ (P × P)
4	3	brel 4591	. . . . . . 7 ⊢ (𝑦<_P 𝑥 → (𝑦 ∈ P ∧ 𝑥 ∈ P))
5	4	simpld 111	. . . . . 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 2553	. . 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 3103	1 ⊢ (𝜑 → 𝐴 ⊆ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∨ wo 697 ∃wex 1468 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ⊆ wss 3071 class class class wbr 3929 Pcnp 7099 <_P cltp 7103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-iltp 7278