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Theorem suplocexprlemss 7935

Description: Lemma for suplocexpr 7945. 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 2579	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<_P 𝑥 → (𝑦 ∈ 𝐴 → 𝑦<_P 𝑥))
3		ltrelpr 7725	. . . . . . . 8 ⊢ <_P ⊆ (P × P)
4	3	brel 4778	. . . . . . 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 2654	. . 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 3233	1 ⊢ (𝜑 → 𝐴 ⊆ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∨ wo 715 ∃wex 1540 ∈ wcel 2202 ∀wral 2510 ∃wrex 2511 ⊆ wss 3200 class class class wbr 4088 Pcnp 7511 <_P cltp 7515

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-br 4089 df-opab 4151 df-xp 4731 df-iltp 7690