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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 4471 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)) | |
2 | prnmadd 10995 | . . . . . . . . 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 1956 | . . . . . . . 8 ⊢ (∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) ↔ (¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑦 +Q 𝑥) ∈ 𝐵)) | |
5 | 3, 4 | sylibr 233 | . . . . . . 7 ⊢ ((¬ 𝑦 ∈ 𝐴 ∧ (𝐵 ∈ P ∧ 𝑦 ∈ 𝐵)) → ∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
6 | 5 | exp32 420 | . . . . . 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 410 | . . . 4 ⊢ (𝐵 ∈ P → ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))) |
9 | 8 | eximdv 1919 | . . 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 | eqabri 2876 | . . . 4 ⊢ (𝑥 ∈ 𝐶 ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
13 | 12 | exbii 1849 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐶 ↔ ∃𝑥∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
14 | n0 4347 | . . 3 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶) | |
15 | excom 2161 | . . 3 ⊢ (∃𝑦∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) ↔ ∃𝑥∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) | |
16 | 13, 14, 15 | 3bitr4i 302 | . 2 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦∃𝑥(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
17 | 10, 16 | imbitrrdi 251 | 1 ⊢ (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → 𝐶 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2708 ≠ wne 2939 ⊊ wpss 3950 ∅c0 4323 (class class class)co 7412 +Q cplq 10853 Pcnp 10857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-oadd 8473 df-omul 8474 df-er 8706 df-ni 10870 df-pli 10871 df-mi 10872 df-lti 10873 df-plpq 10906 df-mpq 10907 df-ltpq 10908 df-enq 10909 df-nq 10910 df-erq 10911 df-plq 10912 df-mq 10913 df-1nq 10914 df-ltnq 10916 df-np 10979 |
This theorem is referenced by: ltexprlem5 11038 |
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