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Theorem suplocexprlemss 7547

Description: Lemma for suplocexpr 7557. 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 2483	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<_P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦<_P 𝑥))
3		ltrelpr 7337	. . . . . . . 8 ⊢ <_P ⊆ (P × P)
4	3	brel 4599	. . . . . . 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 2556	. . 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 3108	1 ⊢ (𝜑 → 𝐴 ⊆ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∨ wo 698 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 ⊆ wss 3076 class class class wbr 3937 Pcnp 7123 <_P cltp 7127

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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-iltp 7302