Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1542 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1542.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
bnj1542.2 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
bnj1542.3 | ⊢ (𝜑 → 𝐹 ≠ 𝐺) |
bnj1542.4 | ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) |
Ref | Expression |
---|---|
bnj1542 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1542.3 | . . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐺) | |
2 | bnj1542.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
3 | bnj1542.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
4 | eqfnfv 6901 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))) | |
5 | 4 | necon3abid 2980 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))) |
6 | df-ne 2944 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐺‘𝑦)) | |
7 | 6 | rexbii 3179 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦)) |
8 | rexnal 3167 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)) | |
9 | 7, 8 | bitri 274 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)) |
10 | 5, 9 | bitr4di 289 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
11 | 2, 3, 10 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
12 | 1, 11 | mpbid 231 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)) |
13 | nfv 1917 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≠ (𝐺‘𝑥) | |
14 | bnj1542.4 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
15 | 14 | nfcii 2891 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
17 | 15, 16 | nffv 6776 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
18 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑦) | |
19 | 17, 18 | nfne 3045 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) ≠ (𝐺‘𝑦) |
20 | fveq2 6766 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
21 | fveq2 6766 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦)) | |
22 | 20, 21 | neeq12d 3005 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
23 | 13, 19, 22 | cbvrexw 3372 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)) |
24 | 12, 23 | sylibr 233 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 Fn wfn 6421 ‘cfv 6426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-fv 6434 |
This theorem is referenced by: bnj1523 33059 |
Copyright terms: Public domain | W3C validator |