Theorem suplocexprlemml 8077

Description: Lemma for suplocexpr 8086. The lower cut of the putative supremum is inhabited. (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
suplocexprlemml	⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))

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

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

Proof of Theorem suplocexprlemml

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
2		suplocexpr.ub	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
3		suplocexpr.loc	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
4	1, 2, 3	suplocexprlemss 8076	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ P)
5	4	sselda 3248	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ P)
6		prop 7836	. . . . 5 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
7		prml 7838	. . . . 5 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
8	5, 6, 7	3syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
9	8	ralrimiva 2623	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
10		r19.2m 3614	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥)) → ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
11	1, 9, 10	syl2anc 415	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
12		suplocexprlemell 8074	. . . 4 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥))
13	12	rexbii 2557	. . 3 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥))
14		rexcom 2715	. . 3 ⊢ (∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
15	13, 14	bitri 184	. 2 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
16	11, 15	sylibr 134	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ⟨cop 3711  ∪ cuni 3933   class class class wbr 4128   “ cima 4775  ‘cfv 5375  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641  Pcnp 7652  <_P cltp 7656

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6368  df-2nd 6369  df-qs 6807  df-ni 7665  df-nqqs 7709  df-inp 7827  df-iltp 7831

This theorem is referenced by:  suplocexprlemex  8083