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Mirrors > Home > MPE Home > Th. List > ltexprlem1 | Structured version Visualization version GIF version |
Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltexprlem.1 | ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} |
Ref | Expression |
---|---|
ltexprlem1 | ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssnel 4416 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)) | |
2 | prnmadd 10407 | . . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ 𝐵) → ∃𝑥(𝑦 +Q 𝑥) ∈ 𝐵) | |
3 | 2 | anim2i 616 | . . . . . . . 8 ⊢ ((¬ 𝑦 ∈ 𝐴 ∧ (𝐵 ∈ P ∧ 𝑦 ∈ 𝐵)) → (¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑦 +Q 𝑥) ∈ 𝐵)) |
4 | 19.42v 1945 | . . . . . . . 8 ⊢ (∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) ↔ (¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑦 +Q 𝑥) ∈ 𝐵)) | |
5 | 3, 4 | sylibr 235 | . . . . . . 7 ⊢ ((¬ 𝑦 ∈ 𝐴 ∧ (𝐵 ∈ P ∧ 𝑦 ∈ 𝐵)) → ∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
6 | 5 | exp32 421 | . . . . . 6 ⊢ (¬ 𝑦 ∈ 𝐴 → (𝐵 ∈ P → (𝑦 ∈ 𝐵 → ∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)))) |
7 | 6 | com3l 89 | . . . . 5 ⊢ (𝐵 ∈ P → (𝑦 ∈ 𝐵 → (¬ 𝑦 ∈ 𝐴 → ∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)))) |
8 | 7 | impd 411 | . . . 4 ⊢ (𝐵 ∈ P → ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))) |
9 | 8 | eximdv 1909 | . . 3 ⊢ (𝐵 ∈ P → (∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑦∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))) |
10 | 1, 9 | syl5 34 | . 2 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → ∃𝑦∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))) |
11 | ltexprlem.1 | . . . . 5 ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} | |
12 | 11 | abeq2i 2945 | . . . 4 ⊢ (𝑥 ∈ 𝐶 ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
13 | 12 | exbii 1839 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 ↔ ∃𝑥∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
14 | n0 4307 | . . 3 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶) | |
15 | excom 2159 | . . 3 ⊢ (∃𝑦∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) ↔ ∃𝑥∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) | |
16 | 13, 14, 15 | 3bitr4i 304 | . 2 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
17 | 10, 16 | syl6ibr 253 | 1 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2796 ≠ wne 3013 ⊊ wpss 3934 ∅c0 4288 (class class class)co 7145 +Q cplq 10265 Pcnp 10269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ni 10282 df-pli 10283 df-mi 10284 df-lti 10285 df-plpq 10318 df-mpq 10319 df-ltpq 10320 df-enq 10321 df-nq 10322 df-erq 10323 df-plq 10324 df-mq 10325 df-1nq 10326 df-ltnq 10328 df-np 10391 |
This theorem is referenced by: ltexprlem5 10450 |
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