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Mirrors > Home > MPE Home > Th. List > Mathboxes > iotaexeu | Structured version Visualization version GIF version |
Description: The iota class exists. This theorem does not require ax-nul 5174 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotaexeu | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval 6298 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
2 | 1 | eqcomd 2804 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
3 | 2 | eximi 1836 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑)) |
4 | eu6 2634 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
5 | isset 3453 | . 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 295 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃!weu 2628 Vcvv 3441 ℩cio 6281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 |
This theorem is referenced by: iotasbc 41123 pm14.18 41132 iotavalb 41134 sbiota1 41138 |
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