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Mirrors > Home > ILE Home > Th. List > acexmidlem1 | GIF version |
Description: Lemma for acexmid 5689. List the cases identified in acexmidlemcase 5685 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
Ref | Expression |
---|---|
acexmidlem.a | ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} |
acexmidlem.b | ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} |
acexmidlem.c | ⊢ 𝐶 = {𝐴, 𝐵} |
Ref | Expression |
---|---|
acexmidlem1 | ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acexmidlem.a | . . 3 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} | |
2 | acexmidlem.b | . . 3 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} | |
3 | acexmidlem.c | . . 3 ⊢ 𝐶 = {𝐴, 𝐵} | |
4 | 1, 2, 3 | acexmidlemcase 5685 | . 2 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}))) |
5 | 1, 2, 3 | acexmidlema 5681 | . . . 4 ⊢ ({∅} ∈ 𝐴 → 𝜑) |
6 | 5 | orcd 690 | . . 3 ⊢ ({∅} ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)) |
7 | 1, 2, 3 | acexmidlemb 5682 | . . . 4 ⊢ (∅ ∈ 𝐵 → 𝜑) |
8 | 7 | orcd 690 | . . 3 ⊢ (∅ ∈ 𝐵 → (𝜑 ∨ ¬ 𝜑)) |
9 | 1, 2, 3 | acexmidlemab 5684 | . . . 4 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ¬ 𝜑) |
10 | 9 | olcd 691 | . . 3 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → (𝜑 ∨ ¬ 𝜑)) |
11 | 6, 8, 10 | 3jaoi 1246 | . 2 ⊢ (({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → (𝜑 ∨ ¬ 𝜑)) |
12 | 4, 11 | syl 14 | 1 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 667 ∨ w3o 926 = wceq 1296 ∈ wcel 1445 ∀wral 2370 ∃wrex 2371 ∃!wreu 2372 {crab 2374 ∅c0 3302 {csn 3466 {cpr 3467 ℩crio 5645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 df-iota 5014 df-riota 5646 |
This theorem is referenced by: acexmidlem2 5687 |
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